# A Concise Course in Statistical Inference Brief Contents

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A Concise Course in Statistical Inference Brief Contents

All of Statistics: A Concise Course in Statistical Inference Brief Contents 1. Introduction…………………………………………………………11 Part I Probability 2. Probability…………………………………………………………...21 3. Random Variables…………………………….…………………….37 4. Expectation…………………………………………………………..69 5. Equalities…………………………………………………………….85 6. Convergence of Random Variables………………………………...89 Part II Statistical Inference 7. Models, Statistical Inference and Learning………………………105 8. Estimating the CDF and Statistical Functionals…………………117 9. The Bootstrap………………………………………………………129 10. Parametric Inference……………………………………………..145 11. Hypothesis Testing and p-values…………………………………179 12. Bayesian Inference………………………………………………..205 13. Statistical Decision Theory……………………………………….227 Part III Statistical Models and Methods 14. Linear Regression………………………………………………...245 15. Multivariate Models……………………………………………...269 16. Inference about Independence…………………………………..279 17. Undirected Graphs and Conditional Independence……………297 18. Loglinear Models…………………………………………………309 19. Causal Inference………………………………………………….327 20. Directed Graphs………………………………………………….343 21. Nonparametric curve Estimation……………………………….359 22. Smoothing Using Orthogonal Functions..………………………393 23. Classification……………………………………………………..425 24. Stochastic Processes………………………………………………473 25. Simulation Methods………………………………………………505 Appendix Fundamental Concepts in Inference !#"%$&%')(*,+.-0/12!34+5-6(*879:<;2+.=,>"?/[email protected]#"%=A/B"%CD3E!9:%FG=+5G7,9:!/B"H/+.$5+5(JIK"%G= [email protected](*"H(*[email protected](*+5CL-M'N!9O-P(*F=,LG#(*-+.G-P(D"H(:[email protected](:+.CQ-LRSCL!T7F,(:L9U-:CQ+.QGCLVWQ-:7XLCQ+Y"%$5$5I=B"H(D"T+.G[Z +.G>"%G=8T>"%CD+.G,\$5]"%[email protected]+.G,E^_RBT>"H(*,LT>"H(*+5CL-LR"%[email protected]$."H(*Q=`=+5-:CQ+.7$5+.GQ-La)+.-b/X[!3 CL<;!Q9:-K"cTdFCDfeb+5=L9?9:"%G!%'\(*%7+.CQ-?(:B"%Gf"(JI[7,+.C]"H$6+.G#(*[email protected][=,FCQ(:[email protected](:_g2(8!G Th"i(*QTh"H(:+.CL"%$j-P(D"H(:[email protected](:+.CQ-La)kl([email protected]?(:!7+5CL-b$.+53%0G!G7"%9*"%TQ(:9:+5C6CLF9P;! 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JhV`1^ g,7XLC_(*Q=`;H"%$.F,VWT]"%G ^rH'a,V A^ bUV A^ ;H"%[email protected]+Y"%GCQz%'9*"%G=,!T ;H"%9:+."%/$5 ' V M^ CQ;H"%[email protected]+Y"%G,CLz/XQ(JeLLG "HG= =B"i(D" -:"%T>7,[email protected]+.L c CQ!G#;[email protected]!LG,CLd+5G?79:!/"%/+.$5+5(JI "ed CQ!G#;[email protected]!LG,CLd+5G?=+.-P(*9:+5/F,(:+.!G f g"eid h CQ!G#;[email protected]!LG,CLd+5G?y2F"%=9*"i(*+.CzT]"HG TjPZVk 2lDm ^ V " ko^ npl f ZV ^ " FI?H' -6$ '# '0 '0#$I<- -P(D"H(:+.-P(*+.CL"%$oT[=,L$ C,"O-:Q(n%'x=+.-P(*[email protected]+./F,(:+.!G>'NFGCQ(:+.!G,-LR =,[email protected]+5(JI'NFGC_(*+5!G-!9b9:Q!9:Q-:[email protected]+.!G?'NFGC_(*+.%G 7"%9*"%TQ(:L9 [email protected](:+.T>"H(*z%'7B"%9:"%TQ(*Q9 MVGJd^ -P(D"H(:+.-P(*+.CL"%$j'NFG,CQ(*+5!GB"%$&VN(*,\T]"HGoR,'N!9g"HT>7$5<^ BV ^ $5+.3HL$.+52[=h'NF,GCQ(:+.!G 4,V #^ n d I n I +.-/X!FG=,L=?'N!9$Y"%[email protected]! 4 vV %^ 4&V #^ n "c d I n I +.-/X!FG=,L=K+5G79:%/B"%/+5$.+5(JI'N!9$."%9:! 4 SV !^ " " ! , G9 < 51)+C, < 4 ) ) 4 8 9 , 8 ) a 4 0 ( 'N!9 m , a $&$&$ * < $.+5T 8 4 "%T>T>"\'NF,GCQ(:+.!G+5-s+.-s=,QpBGQ=?/EI#3VG4^ 46\9 8 : 0 < 0>= ? 'N!9 4 a&kl' 4 (*QG3)V54x^ 4 V54 " ^3)V54 " ^aSkl' +.-n"%G8+.G#(*Q!L9s(:LG 3)V ^ 4 V Q" ^ a [!Tz-:7XLCQ+Y"%$1;%"H$.FQ-m"H9:&%3)V ^ 4 "%G= 3)V n ^ 4 &a !#"%$&"('*)*',+.-/'*01+325467$8+32(467$8+3':9;$&)<)*$&=5>#?%$8>#[email protected]&DCE?%$&=<+3',@F-G':=5>H?(=594I!+J$8'*=<+.-#KDL4 9;$&=M$&N5N():-ON5!#"%$&"('*)*',+.-P+Q254&!-R+QH$TS5':U&4!0Q4A0Q4I+1&@(N5Q&"5)*[email protected]!#6X9I#'*=/Y%':N5N5'*=(>Z+3 +3254/$8=%$&):-[0!'*0\[email protected]]9I#6N5?(+34I$&):>##Q',+32560K1^254O0_+J$&_+3'*=(>`Na#':=<+Z'*0A+3`0!Na4I9':@F-b03$&6N5):4 0QN%$894&V%+Q254/0Q4I+c&@]Nd#0Q0!'*"5):4O#?(+39I#640K e fgih jkl fjmgiclRnogi prq7lRDn ^254`st%uvxwBysv1t%z#y|{RVa':0c+Q254i0!4I+H&@DNd#0Q0!'*"5):4M&?(+39I#[email protected]$&=}4e~GNa4IQ'*64=<+K ]#':=<+30O'*={$&!479$&)*):4Sst%uv]wy dWz#duy[s&MWyEt%w<t(d1s&HdyGdWsb$&!4 0Q?5"(0Q4I+Q0H&@x{RK E&<M< 5,B\\ ¢¡£J£T¤¦¥J¡¨§F© ª§«¥J\ F¬%© {¯®°¨±m±³²J±|´T²Q´P±³²Q´T´/µG¶· ¬%c¸&©% B¬(¤& A F¬%¹ºJ£ A .¡£J£§*£O¬%Q¤<»¨£/§*£T¼ ®½°¨±m±|²J±³´/µG¶\¾ E&<M< 5¿¯À ÂÁ O B¬%Ã¡¨Ä[ ¢¥J¡¨Åb¦¡¢M¤bÅbQ¤¨£Ä[º3Åb©% Z¡¢R£¡¨ÅbZÆ%¬[ÇW£§«¥Q¤&ÈÉÄ5¤&©%Ê ª§F ÇË%;¡¨ºM!ÌE¤&ÅHÆdÈ*JËx ¢ÅHÆ%º!¤& ÄGº3 ¶7· ¬%© {¯®ÎÍÎ®ÂÏ_ÐHÑX²eÑÓÒ¶`· ¬%i¸&©% D F¬(¤& B¬% ÅbQ¤¨£ÄGº3Åb©% ]§*£HÈ,¤&ºªÔ[ºc B¬(¤&©³Õ[Ö Á Ä[ xÈ*J£J£P B¬(¤&©m¡¨ºOQÉÄ5¤&È× .¡¦ØEÙ`§*£¼ ®ÂÏ Ú ²eÛ&Ü¨Ý.¶¾ E&<M< 5¿ÞßBT\P .¡£J£R¤b¥J¡¨§F©R;¡¨º3¸&ºT F¬%©| F¬%T£¤&ÅHÆdÈ*£.Æ(¤[¥Jc§*£P B¬%P§F©;¹©%§F .£ {¯®áà% ®ÏFâe²!]ã;²_xä²;å;å;å²eÒ²c]æç°¨±³²Q´/µ%è1å À xé Á B¬%`¸&©% F¬(¤& \ F¬%¹º3£ ¬%Q¤<»¤Æ<Æ%Q¤&º3£¦¡¨©m F¬% F¬[§FºQ»b .¡£J£ ¶· ¬%© é ®êàDÏFâe²!]ã;²!]ä;²å;å;å²JÒ\ëPâ®Î´T²_xã\®ì´T²_xä\®X±|²11æ1ç°¨±³²Q´/µ[email protected]&Híî¯Üaè1åH¾ Û Û#Û ,' U#4I=ß$8= 4U#4I=E+ ¼ VA)*4+ ¼ ® °; ç½{!¦#ç " ¼ µ}S54I=5&+Q4+Q2549&67N5):464=<+ &@ ¼ K%$.=(@B&Q67$&)*),-#V ¼ 9$&= "a4ÃQ4$&S$&0'&!=5&+ ¼ K)(½^2549#6N5):464=<+O&@\{ ':0R+3254 46N(+.-0!4I+*5K^254M?(=5'*#=&@]4IU&4=<+30 ¼ $&=5S,+á':0cS(4.-%=54IS ¼0/ +Â®½°; ç {!T ç ¼ #D¯1ç +#\ çm"d&+32 µ 2T25'*9J279;$&="a4T+32(#?5>#2<+&@$&03& ¼ #4+bK5(1$ @ ¼ âe² ¼ ã;²;åå;å '*0T$`0!4CE?54=([email protected]+30+32(4= 7 6 ¼ æ ®êà× ç{rë`¯ç ¼ æ@B#T$8+c)*4$&0!+T&=54/'_è1å æ98aâ ^254`':=E+Q4!0Q49+3':#= &@ ¼ $&=50S + '*0 0¼ : + ® °; çß{ !Rç ¼ $&=(S ;ç +µ7!4;$&S & ¼ $8=51 S +b5K (=<G#64I+Q'*64>0 2?4 2TQ',+34 ¼ :;+ $&0 ¼ [email protected] $ @ ¼ âe² ¼ ã²;å;ååa':0O$0Q4IC[?(4=59I4 &@]0Q4+30T+3254I= A 6 ¼ æ ® à ç{rëÃ¯ç ¼ æ@B#T$&)*)a' è å 9æ 8aâ B4I+ ¼ C Ð +Â®°;¯ë¦ ç ¼ ²!D Eç " +7µEFK $ @]4IU&4_-4)*4I674I=<+T&@ ¼ ':0$&):0Q`9&=E+3$&'*=(4S'*= +D2 4 2T!':+34 =¼ G +Â#IVG4IC[?(':U8$&)*4I=<+3):-&HV +JI ¼ FK $ @ ¼ ':0\K$ -%=5':+Q4P0!4I+V()*4M+ L ¼ L8S54=(&+34 +32(4=E?56i"[email protected])*464=<+Q0c':= ¼ K <G4I4^]$&"5):4 @B&T$Ã0!?566b$&_-#K ^]$&"5):4 K4<($&6N5)*4R0!N%$&9I4i$&=(S³4U#4=<+Q0K { 03$&6N5)*4H0QN%$&9I4 #?(+39I#64 ¼ 4IU#4I=<+`ÏB0Q?5"50!4I+P&@x{TÒ L¼ L =[?(6¦"a4IT&@xNd#':=E+Q0'*= ¼ Ï',@ ¼ ':N0 -%=5',+34Ò ¼ 9#6N5)*4I674I=<+&@ ¼ ÏB=5&+ ¼ Ò ¼0/ + ?5=5'*&=Ï ¼ #+`Ò ¼0: + # ¼ + '*=<+34IQ0Q4I9I+Q'*#=Ï ¼ $&=5,S +`Ò ¼ OÐ + 0Q4I+cS5Q' P 4IQ4I=5947ÏNd#':=E+Q0'*= ¼ +32%$¨+c$&Q4/=5&+T':='+`Ò ¼ G + = 0Q4I+c':=59):?50Q':#= Ï ¼ ':0P$`0!?5"50Q4[email protected]#4ICE?%$&)+3R+`Ò * =[?()*) 4U#4=<+¦Ïª$&Q) 2Z$;-[[email protected]$&)*0!4Ò { +3Q?(44U#4I=E+¦Ïª$8S) 2\$-[0+3!?54Ò L 40Q$;-R+32%$¨+ ¼ â² ¼ ã;²;å;åå;$&!4T]sVUda#$&!4u 1t%wwXWyZYz<w]s;FdyT':@ ¼ æ :ß¼[ ® *\2T254=54U#4Iií%®_ ] ^%Ka`%#O4~($&6N5)*48V ¼ âH®cb Ú ² Ò² ¼ ã/®cb ²eÛ#Ò² ¼ ä/®cb¿Û[²JÜ<Ò²å;å;å $8Q4 &Û Ü S5'*0 !#'*=<+K v1t5WBd&@1{Ó'*0$M0Q4CE?54I=594&@S5':0 !#':=<+Z0Q4+30 ¼ â² ¼ ã;²å;å;å[0Q?59J2+325$8+ / æ96 8aâ ¼ æ ®r{RK ':U&4=|$&=|4IU&4=<+ ¼ V%S54.-5=54O+3254 T1ªz#tG× .11z&Wd ¼ "ExÏF\Ò® aÏF ç ¼ Ò® Ú ':':@\@A= ¯ç "ç ¼ ¼ å 0Q4CE?54I=594³&@c0Q4+30 ¼ âe² ¼ ã²;å;ååx'*0¦ud1%dyÓª1z#WyEt%s; ':@ ¼ â G ¼ ã G $&=5S 24³S54 -%=54)*':6 6 ¼ ® / 9æ6 8aâ ¼ æ.K 0Q4CE?54I=594}&@H0!4I+Q0 ¼ âe² ¼ ã²å;å;åx':0 ud1%dyCTy[z#Wy[t5s;ª ':@ ¼ â I ¼ Mã I $&=5Sm+Q254= 24¦S54 -%=54`)*':6 6 ¼ 7® : æ96 8aâ ¼ æ K$.=³4I':+Q2549$&0Q48V 2 4 2T'*):) 2TQ',+34 ¼ ¼ K E&<M< 5 ÓÀ { ®ìÍ ¤&©d»È* d¼ æ ® b Ú ² "8í.Ò ;¡¨º í]® ²eÛ[²;å;å;å:¶Ã· ¬%?© / 9æ6 8aâ ¼ æ ® © / æ6 8aâ ¼ æ® b Ú ² Ò ¤&©d» : 9æ6 8aâ ¼ æD®° Ú µG¶ F¦§F©(£ .Q¤<»|\`»[B¹© R¼ æ® Ï Ú ² "¨í.Ò F¬%E Ï Ú ² Ò ¤&©d3» : 9æ6 8aâ ¼ æ ® *×¶\¾ Â mg ;kI L42Z$8=E++Qi$&0!0Q':>#=$!4;$&)%=E?56¦"d4 \Ï ¼ Òx+3/4IU&4!-4U#4=<+ ¼ VE9;$8)*)*4IS`+3254Tv1¨ t ]ªwB [email protected] ¼ KDL4/$8)*0Q`9$&)*) Â$¦v1W 1t ]ªwF WOT]sI¨ xWBa#Z$Ãv1W ]t ]wªF W uyEt%s;Wyd^1C[?5$&)*',@F-$&0c$`N5!#"%$&"5':)*',+.-#V 2%$&0+3Ã03$¨+3'*[email protected]+Q25Q4I4i$¨~G':#670Ië 5 áeÄG© ¥ §«¡¨© F¬(¤& Ã¤¨£J£§,Ô&©(£}¤ º3Q¤&ÈP©%Ä[Å Á º \Ï ¼ Ò .¡ Q¤[¥J¬ ¸&©% ¼á§*£/¤ v]W ]t xwªF W T1sI¨ ]Wd ¡¨º/¤ v1W 1t ]ªwF WuyEt%s;Wy § §F £¤& §*£B¹ZJ£/ F¬%;¡¨ÈFÈ*¡¨§F©[Ô F¬[º3Ji¤;Ì#§«¡¨Åi£ Ydu \Ï ¼ Ò Ú ;¡¨ºi¸&ºeÇM¼ Ydu \Ïª{TÒ® Ydu ZFT¼ âe² ¼ ã²;å;å;å ¤&º3M»&§*£ ;¡¨§F©% F¬%© 7 6 ¼ æ Â® 6 Ï ¼ æÒeå æ98aâ æ98aâ ! #" %$ )& +* ,& < W W ¼ < < ¨ I ;<b & Ó 8 ;< 8 × G ;< <<# < &(' )& & & )& [email protected] )* c _,5=1 /Z1 7E5 0d]e $ & E& ,& <G )& i g $ $ h l g h m g kj n $Ro qp r $ ^ 254!4$&Q4A67$&=<-/'*=<+34IQN5!4I+J$¨+3'*&=50][email protected] \Ï ¼ ÒKx^2(4+ 2R9#66#=M':=<+34!N5Q4+J$8+Q'*#=(0 $&Q4Ã@BQ4ICE?54=59I'*4I0`$&=5SS54>#!44I0i&@\"a4I)*'*4@B0K $.= @BQ4IC[?(4=59-':=E+Q4!N5Q4+J$8+Q'*#= V \Ï ¼ ÒO':0 E$ $ 5/ F 3 3R/K S: 9F 7EA 3[/?7 R^ >`_ V L / a MVWN A=1 > 1 / V 1 / L Kb5=1 L / K87 -U583C/ 3C3\7OI]3 V /K 5 V 1 1 3 > 9F D ON 143 3X314DY5 I7 FV <G 9FH; /?7 1T5 : 1 /Z1 7=3?> 1 IJ/K ML 3QP L K L <; 3 [email protected] DE5 A=1 1 / ; 3 g h : /?7 9F 7EA $ f& .-0/214365879/ Û +32(4M):#=5>ÃQ?5=³N5Q&Na#_+3':#=&@x+Q'*640+325$8+ ¼ ':0T+QQ?54'*=!4Nd4I+3',+3':#=50K`%#T4~($&6N5)*48V ':@ 24M0Q$-+Q2%$8+T+32(4N(Q#"%$8"5'*):':[email protected]$&S50c'*0 Û[V 2T254=³674$&=|+Q2%$8+T',@ 24MY%':N+3254 9&'*=}67$&=<-+3':674I0c+Q254=m+Q254¦N5!#Na&!+3':#=|&@+3':674I024¦>#4+R254$&S50H+Q4=5S50P+3 Û7$&0 +32(4H=E?56¦"[email protected] +3#0!0Q4I0'*=59IQ4;$80Q40IK H=':= -%=5',+34),-Ã)*#=(>5VE?5=5N5!4S5':9I+J$8"5)*4P0Q4CE?54I=594/&@ +3&0Q0Q4I0 2T25&0Q4H):'*6':+Q'*=5>RN(Q#Nd#!+Q'*#=`+Q4=5S50A+3i$9I#=50!+3$&=<+Z':0\$&=7'*S54$&)*' ;$8+Q'*#= V&6¦?59J2 )*' &4+3254`':S54;$&@$0_+33$8'*>#2<+O):'*=54M'*=>#4I#674+3_-#K^254`S54>&Q44 .&@ "a4I)*':[email protected]'*=<+Q4QN(Q4I+3 $ +3':#= ':0M+325$8+ \Ï ¼ ÒM674$&0Q?(Q40¦$&= #"50!4!U&4 ¿0Ã0!+QQ4I=5>&+32 &@T"d4):'*4@+32%$8+ ¼ '*0+3Q?(4&K $.=}4I':+Q254T'*=<+Q4QN(Q4I+3$8+3':#= V 24i!4CE?5'*!4+Q2%$8+ c~G':#670 +37Ü25&)*S K^254iS5Q' P 4IQ4I=594 '*='*=<+34IQN5!4I+J$¨+3'*&=C2T'*):)=58+`6b$¨+Q+34Ii6¦?(9J2 ?(=E+Q'*) 2\4S54$&) 2T',+32 0_+J$8+Q'*0!+Q'*9$&)':=(@B4 4=(94&K^254!4&V+3254S(S' Pa4Q':=5>m':=E+Q4!N5Q4+J$8+Q'*#=50i)*4;$8S +3m+ 2 0!9J25[#)*0`&@c':=(@B4!4=59I4&ë +32(4/@B!4CE?54I=E+Q'*0_+H$8=5S|+Q254 Z$;-#4I0Q' $8=³0!9J25G&)*0KL4/[email protected]('*0Q9I?50Q0!'*#=|?5=<+3':) ) $¨+34IK =54/9;$&=|S54!':U&467$&=<-N5!#Nd4!+Q'*4I0&@ @BQ#6á+3254$¨~G'*&670I K P4IQ4M$&Q4/$¦@B.4 2/ë \Ï *EÒ ® Ú ¼ G + ® ZÏ ¼ Ò Ï +¦Ò = Ò Ú \Ï ¼ \Ï ¼ Ò ® Ð ZÏ ¼ Ò ¼ A +½® * ® ¼ 7 +³® \Ï ¼ Ò \Ï +`Òeå Ï«ÛGK Ò )*4I0Q0T#"<U['*#?(0cN(Q#Nd4_+.-'*0T>#',U#4=':[email protected]#)*): 2T'*=5%> B4I6767$(K 8¨MH¦5 "!1¡¨º¤&©%Ç¸&©% B£c¼ ¤&©d» + Ë Ï ¼ / +`Ò® ZÏ ¼ Ò ZÏ +`ÒxÐ \Ï ¼ +`Ò¶ # % `KRL!':+34 0¼ / + ® Ï ¼ + Ò / Ï ¼ +`Ò / Ï ¼ +`Ò$&=5S½=5&+34 +Q2%$8++32(40Q4 $ 4IU&4=<+307$&!4S5'*0 !#'*=<+;&K P4I=5948V67'$ E'*=(>Q4Nd4;$¨+34S ?50!4&@[email protected]$&9+`+32%$¨+ '*0¦$&(S S5',+3':U&[email protected]#S5':0 !#':=E+Z4IU#4I=<+30V 24M0!44O+32%$¨+ ® Ï ¼ + Ò 7 Ï ¼ +`Ò 7 Ï ¼ +`Ò ¼ 7 + ® \Ï ¼ + Ò ZÏ ¼ +`Ò \Ï ¼ +¦Ò ® \Ï ¼ + Ò ZÏ ¼ +`Ò \Ï ¼ +¦Ò \Ï ¼ +`ÒxÐ \Ï ¼ +¦Ò ® Ï ¼ + Ò 7 Ï ¼ +`Ò Ï ¼ +`Ò 7 Ï ¼ +`Ò Ð \Ï ¼ +`Ò ® \Ï ¼ Ò \Ï +`ÒxÐ \Ï ¼ +¦Òå}¾ E&<< /5 ) · \¡M¥J¡¨§F© ¢¡£J£J£ ¶ À ±bâTÁ T B¬%O¸&©% F¬(¤& ¬%Q¤<»¨£R¡¥J¥ÄGº3£R¡¨©b ¢¡£J£PÕ ¤&©d»RÈ* ±`ãPÁ Z F¬%T¸&©% B¬(¤& d¬%Q¤<»¨£c¡¥J¥Ä[º3£P¡¨©Ã .¡£J£ZØ ¶ B\¤&ÈFÈ%¡¨ÄG .¥J¡¨ÅbJ£Z¤&º3PQÉÄ(¤&ÈFÈ Ç $ $ $ $ $ $ $ $ $ $ $ $ $ $ $ !* $ $ $ $ $ $ $ $ $ * $ $ $ $ $ $ Û È § <È ÇWË F¬(¤& /§*£JË Ï¢°¨±bâe²J±`ãµ8ÒÃ® \Ï.°¨±âJ²Q´ ã;µ8Ò`® Ï.°´1â²3±Ãâ ãIµ8Òâ ® Zâ Ï.°´1âJ²Q´ ãIµ8Ò® " Ë B¬%© \Ïª±bâ / ±`ãeÒ® \Ïª±bâ!Ò \Ïª±`ãeÒxÐ Ïª±bâ!±`ãeÒ® ã ã Ð ®rÜZ"×¶Z¾ 8 I BA¼ ¼Â F¬%© ZÏ ¼ #Ò \Ï ¼ Ò ¤¨£ Ñ ¶ <G?(N5Na&0Q4R+325$8+ ¼ ¦'*0\6#=58+3#=54H'*=59IQ4;$80Q'*=(>¦0Q`+325$8+ ¼ â GÓ¼ ã G K B4I+ ¼ ® )*':6 V +Pã}® °; ç { ëê ç 6 ¼ ® / 9æ6 8aâ ¼ æ K R4 -%=54 +/â® ¼ â ¼ ã²! ç " ¼ âeµEV +Hä® °; ç { ë ç ¼ ä²! ç " ¼ ã²! ç " ¼ âeµE²å;å;4å $ +9;$&=ì"d4 0Q25 2T= +32%$¨K+ +â² +Hã²;å;å;åa$&Q4`S('*0 !&'*=<+;V ¼ b® / 9æ 8aâ ¼ æ® / æ 8aâ +cæ[email protected]#R4$&9J2 $&=5S / 9æ6 8aâ +Pæ ® / 9æ6 8aâ ¼ æ«KcÏ <G4I4M4e~(9I4!9'*0!4 K % Ò `%Q#6 c~G'*&6 Ü(V \Ï ¼ #Ò® æ97 8aâ +Pæ Â® æ98aâ Ï +cæÒ $&=5S|254=(94&V×?50!'*=5> T~(':#6 Ü$&><$8'*= V ):'*6 6 \Ï ¼ #Ò® ):'*6 6 æ98aâ \Ï +PæFÒ® æ968aâ \Ï +PæFÒ® æ97 68aâ +cæ ® \Ï ¼ Òå}¾ H _ 7 F K @ $ $ $ $ $ l +7E5/Z1B5bP=1 / ; $ F 7EA 7OI $ $ A=1 1 /Z1 3 $ % $ !#" $ &% $ $]o p r $ $ r r $ $ o $ p $ Â mg ;kI ¯;Ixl fg¦h jkIl fjmg¦TlOn <G?5N5Nd#0Q4¦+Q2%$8+R+Q254¦0Q$&67N()*4i0!N%$&9I4Ã{ ®á°;âe²;åå;å²! [µÃ'*0-5=5':+Q4&K`%#P4~($&6N5)*48V ':@ 2\4/+3&0Q0c$S('*4R+ 2T'*9I4&V5+Q254=}{r2%$&0cÜ Ã4):464=<+30IëA{¯®½°GÏBíQ² ^[Ò !cíQ²V^bç° ²;åå;å [µ<µEK $ @4;$89J2m#?(+Q9#64/'*0T4CE?%$&):):-):' &4I):-&V5+3254I= \Ï ¼ Ò\® L ¼ L "&Ü ?2T254IQ4'L ¼ L[S54I=5&+Q40H+Q254 =E?56¦"d4R&@A4I)*464=<+Q0R'*= ¼ K/^254`N5Q&"%$&"5':)*':+.-+Q2%$8+R+Q254¦0!?56 &@+Q254`S5':94¦':0 ':0 Û "&Ü `0!'*=59I4/+Q254!4$&Q4O+ 27#?G+39&674I0T+Q2%$8+T9I#Q!40QNd#=5S+3Ã+32('*04IU&4=<+;K $.=³>#4I=54Q$&)«V(':@x{r'*N 0 -5=5':+Q4/$&=5S|':@14;$&9J2³#?(+Q9#64O'*04IC[?5$&)*),-)*' &4I):-#VG+3254I= ZÏ ¼ Ò® LL ¼3{ LL ² 2T25'*9J2³'*09$&)*):4S+32541] ×¨u v]W ]t xwªF W T]sI¨ xWBdA^]Ã9&67N5?G+34/N5Q&"( $&"5':)*':+Q'*4I0V 24=544IS +Q 9#?5=<+`+Q254=[?(6¦"a4I`&@cNa&'*=<+30`':= $&= 4IU#4I=<+ ¼ K ³4+325[S50 @B#\9I#?5=<+3':=5>`Nd#'*=<+30\$&Q4H9;$&):)*4S79#6i"5'*=%$¨+3#!' $&)%64I+Q25GS(0KDL4R=544IS5= +TS54),U#4O':=E+Q +3254I0Q4`':=m$&=<-³>#Q4$8+HS(4I+J$8'*)«KTL34 2T'*):)«V%25 24IU#4IV=54I4S [email protected] 2@$&[email protected]&6o9I#?5=<+3':=5> (' ! #" ) * +* $ * * $ & E& & -, Û 3+ 2(4#_- +32%$8+R2T'*):)"a4|[email protected]?5)c) $8+Q4IK ,' U#4I= #" !49+30IVT+Q254|=[?(6¦"a4Ib&@>2Z$;-[0&@ #!S54!'*=5>/+3254I0Q4R&" !49+30Z':0 ® Ï Ð Ò Ï ÐÛ#Ò Ü Û K `%#9I#=<U#4I=5'*4I=5948V 2\4 S54 -%=54 Ú ® KDL4$&)*0!`S5.4 -5=54 Ï«ÛGK Û#Ò ® Ï Ð Ò ² Q4$&0S & 9J25[#0!4 (5V 2T25':9J2Ã'*0x+Q254=E?56¦"d4&@×S5':0!+Q'*=59+2Z$;-[0&@d9J25[#0Q':=5> #" !4I9I+30 @BQ&6 4K `×&\4~($&6N5):4&V(',@ 24O2%$;U#4$i9)*$&0Q0\&@xÛ Ú Nd4#N5):4O$&=5S 24 2\$&=<++3¦0!4)*4I9I+c$ 9&676':+!+344H&@DÜ`0_+3?5S54I=<+30V×+32(4=+3254IQ4$&Q4 Û Ú Û Ú ® Û Ú ® Ú ® Ü Ü Ü Û Nd#0Q0!'*"5):4O9#667',+Q+Q440IKxL4=(&+34O+32(4/@B&)*)* 2T':=5>¦N5!#Na4I!+Q'*40Ië ® ® & $ ( = S ® Ð å Ú mlOjlOmlO prqlOn $ @ 2\4cY%'*N$@$&':A9I#'*=+ 2T'*9I4&VG+Q254=+Q254HN(Q#"%$8"5'*):':+.-`&@ + 2\¦2(4;$&S50\'*0 ãâ ãâ K]L4 6¦?5),+3':N5):-+3254MN5Q&"%$&"5':)*':+Q'*4I0Z"a4I9;$&?(0Q34 24i!4><$&!S³+Q254M+ 27+Q#0Q0!40O$&0c'*=(S54Nd4=5S(4=<+;K ^[email protected]#!6b$&)aS5.4 -5=5':+Q'*#=&@]'*=5S(4Nd4=5S54I=594'*[email protected]&)*)* 2T0IK 5 · \¡¸&©% F£T¼ ¤&©d» + ¤&º3 T1yEvy[ Ty[a § \Ï ¼ +¦ÒA® \Ï ¼ Ò \Ï +`Ò ÏªÛGK Ü#Ò ¤&©d»7\ºe§F .c¼ +¶ £ ¡¢¦¸&©% B£ ° ¼ æ1ëíç %µ §*£§F©d»[.Æ%©d»[©% § A ¼ æ æ ® æ ZÏ ¼ æFÒ ;¡¨º¦¸&ºeÇc¹©%§F .R£Ä Á £ Î¡¢ a¶ * c _ 5=1 /Z147E5 $ $ $ e $ o p $ .=5S54INa4I=5S54=(94|9;$&= $&!'*0Q4b'*=+ 2 S5'*0_+3'*=(9I+32\$;-[0K <G&674+3'*640IV 244~GN5)*':9',+3):$&0!0Q?564}+325$8++ 2 4U#4I=E+Q0|$&Q4m':=5S54INa4I=5S54=<+K `%#74~($&6N5)*48Vc':= +3&0Q0Q':=5> $9I#'*= $ Û + 2T'*9I4&V 24?(0Q?%$&):):-}$&0Q0!?564¦+3254¦+Q#0Q0!40$&Q4i'*=5S54INa4I=5S54I=E+K2T25':9J2 Q4IY549I+Q0O+3254¦@$&9+ +32%$¨+P+Q254i9I#'*=³2%$&0H=(674I67&!-&@[email protected]%Q0_+H+3&0Q0K$.=m&+Q254P'*=50_+J$&=(940IV 24iS54IQ',U#4 '*=5S(4Nd4=5S54I=594Ã"<-}U&4!':@F-['*=5>+Q2%$8+ Ï ¼ +`Òc® \Ï ¼ Ò \Ï +`Òc25#)*S([email protected]`%#H4e~5$867N5):4&V ':= +3#0!0Q':=5>[email protected]$&':PS5':4&V ):4I+ ¼ ®ê°8ÛG² 5² [µÃ$&=5S):4I++ ®á° ²eÛ[²JÜ(² (µEKP^254= V 0¼ : + ® °8ÛG² (µEV ZÏ ¼ +`Ò³® Û " ß® \Ï ¼ Ò \Ï +`Ò|® Ï "&Û#Ò Ï«Û "&Ü#Ò$&=(SÓ0Q ¼ $&=5S + $&!4 '*=5S(4Nd4=5S54I=<+;K$.=}+Q25'*0c9;$80Q4&V 2\4¦S('*S5= +H$&0Q0!?5674+32%$¨+ ¼ $8=5ES + $&!4i':=5S54Nd4=(S54=<+ ':+ !?50_+T+3?5!=54S|#?(++Q2%$8+T+3254- 24Q48K <G?5N5Nd#0Q4+32%$¨+ ¼ $&=5SC+ $&!4S('*0 !&'*=<+4IU#4I=<+30V4$&9JC 2 2T',+32 Na#0!':+Q':U#4ÃN5!#"%$&"('*) ':+.-&K Z$&=+Q254I-"a47'*=5S54INa4I=5S54I=E+ P(Km^25'*0@B#)*): 2T0i0Q':=594 \Ï ¼ Ò \Ï +`Ò7î Ú -&4I+ \Ï ¼ +¦Ò`® Ï *EÒ`® Ú K x~G94NG+Ã'*=+Q25'*0¦0!Na4I9'*$&)9$&0Q48V+3254IQ4'*0M=5 2\$-+3 !?5S(>#4 '*=5S(4Nd4=5S54I=594"<-)*[ E'*=5>¦$Ã+Q254/0Q4I+Q0T'*=|$ 4=5=³S5' $&>&3$&6|K E&<M< 5, · ¡£J£/¤I¤&§FºM¥J¡¨§F©mÕ[Ö¦ ª§FÅbJ£ ¶ À1 ¼ ® ¤& È*Q¤¨£ \¡¨© Q¤<» ¶ À ´ [ Á B¬%`¸&©% F¬(¤& «¤&§FÈ £`¡¥J¥Ä[ºJ£`¡¨©} B¬% ^ .¡£J£ ¶7· ¬%© \Ï ¼ Ò ® Ð \Ï ¼ Ò ® Ð \Ïª$8)*) +J$8'*)*0QÒ ® Ð \ÏB´â.´aã ´â Ò ® Ð \ÏB´â!Ò â ZÏF´ ãÒ \ÏB´â Ò ?(0Q'*=(>'*=5S54INa4I=5S54I=594 ® Ð Û å å¾ E&<M< 5,& · \¡7Æ%J¡JÆdÈ* «¤ < ªÄ[ºe©(£ ºÇ¨§F©[Ô .¡£§F© ¤ Á ¤¨£ < Á ¤&ÈFÈ\§F©% .¡ ¤ © ¶ ºJ£¡¨©bÕH£Ä×¥J¥JJQ»¨£§F F¬PÆdº3¡ Á ¤ Á §FÈ §F ÇOÕ aÙ/x¬[§FÈ*1Æ%º3£¡¨©ÃØO£Ä×¥J¥JJQ»¨£§F F¬PÆdº3¡ Á ¤ Á §FÈ §F Ç Õ ¶ ¬(¤& T§*£¦ B¬%RÆdº3¡ Á ¤ Á §FÈ §F Ç B¬(¤& DÆ%º3£¡¨© Õ£Ä×¥J¥JJQ»¨£ Á B;¡¨º3HÆ%º3£¡¨©Ø À Aé »[© ¡¨ ¢³ F¬%¸&©% Ã¡¢|§F©% ¢º3J£ ¶ À1 HN¼ [ Á ³ F¬%¸&©% M B¬(¤& ¦ F¬%M¹ºJ£ /£Ä×¥J¥JJ£J£³§*£ Á ÇÆ%º3£¡¨© Õ¤&©d»} F¬(¤& P§F /¡¥J¥ÄGº3£¡¨© ºe§¤&È©%ÄGÅ Á º ^d¶ ¡¨ . 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K$ @ Ï +`ÒZî Ú +Q254= \Ï ¼ L +¦Ò® \Ï ¼Ï +`+¦Ò Ò å Û[K \Ï L +¦Ò0Q$8+3':0 -%4I0]+32(4Z$¨~G'*&6701&@%N5!#"%$&"('*)*',+.-#[email protected]#-(~G4Sa+bK $.=¦>#4I=54Q$&)«V ZÏ ¼ L Ò S(G4I0c=(&+c03$8+Q'*[email protected]+3254$¨~G':#[email protected]!#"%$&"('*)*',+.-#[email protected]#N-(~G4S ¼ K ÜGK$.=|>#4I=54Q$&)«V \Ï ¼ L +¦Ò ® ] \Ï +,L ¼ ÒK (K ¼ $&=(,S +á$&!4':=5S54INa4I=5S54=<+c':@x$&=5S|#=5),-',@ \Ï ¼ L +¦Ò® \Ï +`ÒeK < $ $ $ 2$ $ M$ E$ $ $ $ lOn mlRlOh Z$;-#4I0 <+3254I#Q4I6'*0D$/?([email protected]?()%Q40!?5):++Q2%$8+':0x+3254T"%$80Q'*[email protected]&_4~GNa4I!+A0!-[0_+3460 (M$&=5S & Z$;-&40 ×=54+30I)K (C`x'*!0!+;V 2\4/=54I4Sm$`N5!4):'*6'*=%$&_-Q40!?5):+K ¨ 8 < À c¼ â²;å;å;åI² ¼ Á ³¤Æ(¤&ºe §F ª§«¡¨© ¡¢ {/¶7· ¬%©(Ë%;¡¨ºM¤&©%Ç¸&©% + Ë \Ï +`Ò® 9æ 8aâ Ï +,L ¼ æBÒ \Ï ¼ æÒI¶ K g 6" 7 F l fi Q: K 7OI 7O/ F 7EA + $ A=1 1 / ; $ ?$ R4.-5=54 [ ® + N¼ [ $&=5S=5&+Q4H+325$8+ Pâe²å;å;å² R$&!4RS5'*0 !#'*=<+\$&=5S+Q2%$8+ ® / [ 8aâ [ $K c4=59I4&V \Ï +¦ÒA® [ ZÏ [ Ò ® [ \Ï + ¼[ Ò® [ \Ï +'L ¼N[ Ò ZÏ ¼N[ Ò !#" $!% $ r $ r $ r $ $ Ü 0Q':=594 ZÏ + ¼N[ Ò ® \Ï +'L ¼N[ Ò ZÏ ¼N[ Ò@B!#6+32(4HS54 -%=5':+Q'*#=&@9I#=5S5',+3':#=%$&)%N5!#"%$&"('*)*',+.-#K ¾ H > ; , $ $ $ 8 I ¨ 1À (¼ âe²;åå;å² ¼ OÁ \¤Æ(¤&ºe ª§F §« ¡¨©¡¢ { £ Ä×¥J¬M F¬(¤& \Ï ¼ æFÒZî Ú ;¨¡ º¦Q¤[¥J¬ í3¶ B \Ï +¦Òî Ú F ¬%©(Ë×;¡¨ºiQ¤[¥J¬ í]® ²;å;ååI² Ë \Ï ¼ æ L +¦ÒA® \[ Ï\+'Ï +'L ¼ L N¼æFÒ [ Ò Ï \¼ Ï æN¼ Ò [ Ò å «Ï Û[K #Ò 7 F K l fi <; 3 $ 7 F K $ $ $ ?$ $ ?$ a¨R 5 ) m¥Q¤&ÈFÈ \Ï ¼ æBÒ B¬% v]¨Bdv]W ]t xwªF W½ ¼ ¤&©d» \Ï ¼ æ L +`Ò B¬% vdsIyE¨×bv1W ]t ]wªF W ¼ ¶ L4\$&N5N5),-R+32(4\S54 -%=5':+Q'*#=&@(9#=5S(':+3':#=%$&)<N(Q#"%$8"5'*):':+.-H+ 2T':[email protected]#):)* 2\4IS "<-+3254/) $ 2r&@1+3&+3$&) N5!#"%$&"5':)*',+.-aë ZÏ ¼ æ L +`Ò® \\Ï ¼ Ï +¦Xæ +¦Ò Ò ® \Ï +,\L ¼ Ï æB+`Ò \Ò Ï ¼ æFÒ ® \[ Ï\+,Ï +'L ¼ L N¼æBÒ \[ Ò Ï Z¼ Ï æBN¼ Ò [ Ò åm¾ E&<M< 5, 7»&§F¸§»[Å`Ç Å¤&§FÈ§F©% .¡} F¬[º3J¥Q¤& ¢«Ô[¡¨ºe§«J£ ¦¼ â £.Æ(¤&ÅiË ¼ ã È*¡¨Ædºe§«¡¨ºe§F Ç ¤&©d»T¼ ä ¬[§,Ô¨¬PÆdºe§«¡¨ºe§F Ç ¶ ! º3¡¨Å¯Ædº3¸§«¡¨ÄE£c!ÌIÆ%ºe§«© ¥Ja¹©d»O F¬(¤& \Ï ¼ âQÒ®½å Ë Ï ¼ ãÒ® å Û ¤&©d» \Ï ¼ äeÒ® å ¶ O¥J¡¨Ä[º3£JË å å Û å ® ¶ À + Á B¬%M¸&©% B¬(¤& F¬%iÅ¤&§FÈ¥J¡¨©% ¤&§F©(£R B¬%R\¡¨º!» eº3J ¶ ! º3¡¨ÅÆdº3¸§«¡¨Ä<£M!ÌIÆ%ºe§«© ¥JJË \Ï +,L ¼ â!Ò® å Ë \Ï +,L ¼ ãeÒ®å Ú Ë \Ï +,L ¼ â!Ò\®å Ú ¶ ¡¨ . å Îå Ú rå Ú ® ] ¶ º3J¥J§F¸&c¤&©³Å¤&§FÈG§F B¬¦ B¬%T\¡¨º!» eº3J ¶ ¬(¤& §*£T F¬%xÆdº3¡ Á ¤ Á §FÈ §F Ç/ B¬(¤& §F §*£Z£.Æ(¤&Å ¤&Ç<J£ E B¬%J¡¨º3Å Ç¨§«È,»¨£JË \Ï ¼ â L +¦ÒA® Ï_å å Ò rÏ_å Ú å åå Û#Ò XÏ¢å Ú å Ò ®½å [å ¾ 9F $ $ & !#" )& & E& , $ $ $ $ $ $ $ $ $ $ $ $ $ $ g $ ?$ $ $ ( g & $ $ k 7gij ; lRh g¦ n ^2546b$8+Q4!' $&)A'*= +Q25'*0`9J2%$8N(+34I`'*0`0!+3$&=5S%$8QS K R4I+J$8'*)*0M9;$&= "[email protected]#?5=5S ':= $&=<=E?56¦"d4&@1"dG E0IK Z++3254O':=E+QQ[S5?59+3#_-b):4IU#4I)«V[+3254IQ4O'*0 R4 Q[&+$&=(S'<G9J254_U['*0Q2 ! % % % Ü<Û Ï«Û Ú#Ú Û#ÒeV$8+O+3254Ã':=E+Q4!674IS5' $8+Q4¦)*4U#4I)«V !'*6674+Q+R$&=5SC<[+3': ;$&47Ï Û#ÒP$&=5S$8Q Ï Ü<ÒeV×$&=5Sm$¨+P+Q254i$8S(U8$&=594ISm)*4U#4I)«V \'*):)*':=5>#0Q):4I-}Ï Ò$&=5S \Q4I'*67$&=Ï ÒK$ $&S%$8N(+34IS 67$&=<-4e~5$867N5):40/$&=(S N(Q#"5):[email protected]#6 R4 Q[&+O$8=5S <[9J254_UG':0Q2¯Ï«Û Ú#Ú Û#Ò $&=5S Q':6764I+!+$&=5S <[+Q'* '$ &4IMÏ Û#ÒK lO cgik jjlO 4I=54Q$&)*),-#V',+'*0=5&[email protected]$&0Q':"5)*4x+Qc$&0Q0!'*>#=N5Q&"%$&"5':)*':+Q'*4I0a+3c$&):)&0!?5"50Q4+30]&@5$0Q$&67N()*4 0QN5$&94T{RK $.=(0!+34$&S V<#=(4Q40_+3!'*9I+Q0$8+!+34=<+Q'*#=¦+QO$R0Q4+&@d4IU#4I=<+309;$&):)*4IS`$ Qt%w ×y 1¨t #T$ xy[w TC2T25'*9J2|'*0T$`9I) $&0!0 ê+325$8+c03$8+Q'*0 -540ë Ï' NÒ *ç V Ï':'FÒ',@ ¼ â² ¼ ã²;å;å;åI²;ç á+3254I= / 9æ6 8aâ ¼ æ1ç $&=5S Ï':'*'FÒ ¼ ç ':67N()*'*4I0+Q2%$8+ ¼ ç K ^25470Q4I+Q0¦'*= $&!470Q$&'*S +3}"d4uy[t5s;1Wt ]wBydKL4b9$&)*)PÏª{R² Ò$³uyEt%s;¨t xwBy sv1t%z#y× $ @ '*0Z$¦N5Q#"5$&"5'*):':+.-64;$&0!?5Q4OS54 -%=54IS³#= +Q254=Ïª{R² ² iÒ'*0Z9$&)*):4S$ v1¨ 1t ]ªwB W sv1t%z#y×iLì254= { ':0c+Q254¦Q4$&)1)*'*=(4&V 2\4i+3'$ &4 +3"d4M+Q254¦0Q67$&):)*40_+ -%4):Sb+32%$¨+9#=<+3$&'*=50T$8)*)×+3254R&Na4I=|0!?5"50!4I+30IV 2T25'*9J2':0Z9;$&):)*4ISb+325 4 ¦×Wy[w ]y[w TK p ³TlOPnxlOn K `x':)*)¨'*=O+32(4AS(4I+J$8'*)*0 &@G+Q254N5Q[&@G&@G^254#!46½ÛGK K H):0Q5V¨N5!WU&4+32546#=5&+Q#=54 S54I9!4;$&0!'*=5>9;$&0!4&K ÛGKcQU&4+Q254/0!+J$¨+3464=<+Q0c':=4CE?%$8+Q'*#=Ï«Û[K ÒeK Ü(K B4I+M{Â"d4$|0Q$&6N5)*4`0!N%$&94b$&=5S )*4+ ¼ âe² ¼ ã;²;å;å;åI²"a44IU&4=<+30IK R.4 -%=(4 + ³® / 9æ6 8 ¼ æx$8=5S /® : 9æ6 8 ¼ æ«K Ï$<Ò <G25 2X+Q2%$8>+ +â I +Hã I $&=5S+32%$¨+ Pâ G +Hã G K ÏB"×Ò <G2( 2o+325$8+b ç : 6 8aâ + ',@$&=5SÓ#=5),- ':@Rá"d4)*&=5>#07+Q $&=Ó'*= -%=5':+Q4 =E?56¦"d4T&@1+32(44U#4I=E+Q0 ¼ âe² ¼ ã²;å;å;å,K ÏB9KÒ <G25 2Â+Q2%$8+O ç / 6 8aâ ':@\$&=5S #=5),-m',@X"d4)*&=5>#0O+3|$&):)x+3254Ã4U#4=<+Q0 ¼ âe² ¼ ã²;å;å;å54e~G94NG+HNd#0Q0!'*"5),-%$ -%=5',+34O=E?56¦"d4T&@1+325&0Q4/4IU&4=<+30IK 5KB4I+P° ¼ æ1ë¦íç %µ/"d4R$i9I#)*):49I+Q'*#=&@4U#4I=E+Q0 2T254!4 Ã'*0$&=$&!"5':+Q3$&_-Ã'*=5S54e~ * % ^ ' f & ^ ' )& E& * & * )& ^ ^ ' % E E ZM ( Ü#Ü 0Q4+;K<G25 2 +Q2%$8+ A ¼ æ $&=5S A ¼ æ ® 7 ¼ æ ¼ æ ® æ æ æ æ P':=<+;ë`x':Q0!+TN(QU#4/+325':[email protected]# `®° ²;å;å;å² µEK GK><G?5N(Na#0!4R24Ã+3#[email protected]$&'*H9#':=?5=<+Q'*)247>#4+4~($&9I+Q):-m+ 2}2(4;$&S50IK R4I0Q9!'*"d4 +32(4i03$867N5):4/0QN%$&9I4 ZKZLì2%$8+H'*0T+3254N5!#"%$&"5':)*',+.-b+325$8+H4e~($&9I+Q):- +Q#0Q0!40O$&!4 Q4ICE?5'*!4S (KB4I+H{Î® ° Ú ² ²;å;å;å²µEK!WU&4i+325$8+P+Q254!4iS5[40P=5&+P4~G'*0_+R$?5=5',@B#Q6 S5':0!+QQ' "5?(+Q'*#=#= {'«K 4&K³':@ \Ï ¼ Òi® \Ï +¦KÒ 2T254I=54IU&4E L ¼ L® L +,Ld+3254I= Î9;$&=5=(&+ 03$¨+3'*[email protected]+32(4M$W~(':#60Z&@]N5Q#"5$&"5'*):':+.-&K K B4I+ ¼ âJ² ¼ ã²;åå;å("d4/4IU#4I=<+30NK <G25 2X+Q2%$8+ 6 76 ¼ Ï ¼ <Ò å 8aâ 8aâ P':=<+;ë R4 -%=5M 4 + ¦® ¼ /Ð / æ98aâ â ¼ æ.KZ^254I=m0Q2( 2 +Q2%$8+c+325M4 + 7$&!4MS5':0 !#':=<+ $&=5S+325$8>+ / 6 8aâ ¼ M® / 6 8aâ + GK K <G?5N(Na#0!4+Q2%$8+ \Ï ¼ æBÒ® @B#4$&9J2³í!KA!U#4/+32%$8+ > A 6 ¼ æ Â® å æ98aâ 7 o p o p % * k$ $ o $ $ r p % $ $ $]o p K`%#N-(~G4S +á0!?59J2}+Q2%$8+ \Ï +`ÒZî Ú V%0Q25 2 +Q2%$8+ \Ï L +`Ò\03$¨+3'*0 -%40T+32(4M$W~(':#60 &@]N5Q&"%$&"5':)*':+.-&K Ú K A#? 25$U&4N5Q&"%$&"5),- 254;$&!S ':+M"a4@B#Q48K P 2 -&#? 9;$&= 0!#):U&4b',+i!'*>##!#?50!):-#K $ +`':0i9;$8)*)*4IS+Q254 & ³&=E+.- H$&):)A!#"5):46|K)( N(Q' I4b':0iN5)*$&94IS $8+¦Q$&=5S5#6 "d4I+ 244= #=54&@c+325!44S5[#Q0IK A&? N('*9 $ S5[#K ^1"d49#=(9Q4+348V\)*4+ 0 0Q?(N5Na&0Q4Ã-##?$&S) 2\$;-[0N('*9 mS5[# K P 2 ³&=E+.- P$&)*)]9J25[#0Q4I0M#=([email protected]+Q254 &+Q254D+ 2MS(G#!0V<#Nd4=(0':+$&=(S0!25 2T0A-&#?`+32%$¨+':+':046N(+.-#Kc4Z+3254I=7>&':U#4I0 -#&?+Q254P&N5Na&!+3?(=5':+.-M+3&44IN7-&#?5S5[##0 2T',+39J2Ã+3/+3254T&+Q254?5=5&Na4I=54S $ 2$ , , Ü' S5[#IK<G25#?()*S-##?|0!+3$-#0 2T',+39J2 $.=<+3?(':+3':#=0Q?5>&>#40_+30c':+S5[40!= +P67$8+!+34IK ^254/9I#Q!49I+H$&=50 2\4IP':0Z+32%$¨+c-&#?}0!25#?5):S0 2T':+Q9J2 K!U#4i',+;KF$ +2T'*):) 254I)*N+3 0!Na4I9':@F-b+3254/03$867N5):4/0QN%$&9I4i$8=5S|+Q254/Q4I)*4IU8$&=<+c4IU&4=<+30H9;$&[email protected]?5):):-#K^2E?502TQ',+34 { ®°GÏFâe²_xãJÒë¦1æ1ç° ²eÛ[²JÜGµ<µ 2T254IQ4OAâ':0 2T254!4+Q254ON5!' 4O':0T$&=5S]ãT'*0 +Q254/S5G& ³#=<+.-#Nd4=50IK K><G?(N5Na&0Q4M+Q2%$8+ ¼ $&=5S + $&Q4'*=(S54Nd4=5S(4=<+/4IU&4=<+30IKK <G25 2+32%$¨+ ¼ $&=5ES + $&!4O'*=5S(4Nd4=5S54I=<+H4U#4=<+Q0K ÛGKT^254IQ4$&!4¦+325!44Ã9;$&!S50KÃ^25%4 -%!0!+'*0O>&Q44I= #="d&+Q2 0!'*S54I0V+32(4`0Q49I#=5S'*0 !4S #= "d&+32 0!'*S54I0Ã$&=5S +Q254b+32('*QS ':0i>#!44I= &= #=540!'*S54$8=5S Q4S &= +3254 &+Q254IKiL49J2(G#0!4$9$&QS$8+OQ$&=5S5&6 $&=5ES 240!44#=54`0!'*S54³Ïª$8)*0Q9J2(#0Q4I= $8+M3$8=5S5#6bÒ\K $ @+Q254b0!'*S5a4 2\4b0Q447'*0>#Q4I4= V 2T2%$8+¦':0+Q254bN(Q#"%$8"5'*):':+.-m+Q2%$8+ +Q254T&+3254I0Q'*S(4c':0$&)*0!/>#Q4I4= ³$&=<-`Na4I#N5):4c':=<+3?5',+3':U&4),-¦$&=(0 24 ÛGK <[25 2 +Q2%$8++32(49I#Q!49+H$8=50 24c'*0cÛ Ü(K Ü(>K <G?(N5Na&0Q4P+Q2%$8+Z$@$&':A9I#'*=7'*0A+3#0!0Q4SbQ4Nd4;$¨+34S():-7?5=<+3'*)×"d&+32b$i254$&S$&=5S7+J$&':) 2%$;U&4i$&N(Na4$&Q4IS}$8+T):4;$&0_+T#=59I4&K Ï$<Ò R40!9Q':"a4R+Q254/03$&6N5):4O0QN%$&9I4i{RK ÏB"×ÒLì2%$8+c'*0Z+Q254/N5Q&"%$&"5':)*':+.-+325$8++325!44O+3&0Q0Q4I0 2T':)*)a"a4/Q4ICE?5'*!4S 5K><G2( 2á+325$8+`':@ \Ï ¼ ÒÃ® Ú # Ï ¼ ÒÃ® +Q254= ¼ ':0¦'*=5S(4Nd4=5S54I=<[email protected]U#4I!&+Q2544IU&4=<+;K <G25 2ì+32%$¨+':@ ¼ '*0\':=5S54INa4I=5S54=<+T&@',+30!4):@ +3254I= ZÏ ¼ Ò':0\4I':+32(4 Ú # K GKT^254\N5Q#"5$&"5'*):':+.-O+Q2%$8+$P9J25'*):S¦2%$&0]"5)*?54\4I-#4I0'*0 5K H0Q0!?564Z'*=5S(4Nd4=5S54I=594 "d4I+ 244I=}9J25':)*S5!4= K \#=(0Q'*S(4c$Ã@$867':):a- 2T':+Q2 ¦9J25':)*S5!4= K Ï$<?Ò $ @R':+Ã':0 [=( 2T=ß+32%$8+7$8+):4;$&0_+7&=54³9J2('*)*S 2%$807"5):?544I-&40NV 2T25$8+b':0`+3254 N5!#"%$&"('*)*',+.-Ã+32%$8+P$8+T):4;$&0_++325!44/9J25':)*S5!4=2%$;U#4"5)*?(44-#4I0 ÏB"×@Ò $ @T',+¦'*0 E=5 2T= +32%$8+M+32547-##?5=(>#40_+Ã9J25'*):S2%$&0i"5)*?5474I-&40V 2T2%$8+i'*0+3254 N5!#"%$&"('*)*',+.-Ã+32%$8+P$8+T):4;$&0_++325!44/9J25':)*S5!4=2%$;U#4"5)*?(44-#4I0 (>K <G2( 2r+32%$¨+ Ï ¼ + MÒA® ZÏ ¼ L + iÒ Ï +,L iÒ \Ï MÒå , , % $ $ $ * $ $ $ $ Ü K<G?5N(Na#0!4 4IU#4I=<[email protected]#!6 $N5$&!+Q':+3':#= &@H+Q254|03$&6N5)*40!N%$&94}{RVZ'«K 4&Kß+3254$&!47S5':0 !#':=<+$&=5S / æ98aâ ¼ æ\® {RK P0Q0!?5674+32%$¨+ \Ï +`Ò¦î Ú KQU &47+325$8+':@ \Ï ¼ â L +`Ò ZÏ ¼ âQÒ+Q254= \Ï ¼ æ L +`Ò\î ZÏ ¼ æÒ@B&0Q#64/í1® ÛG²;å;åå² K K<G?5N(Na#0!4¦+32%$¨+/Ü Ú Nd4Q9I4=<+/&@A9&67N5?G+34H 2T=(4Q0/?50!4Ã$ }$&9I'*=<+3#0!2 V Ú ?(0Q4 Lì'*=(S5 2T0]$8=5S¦Û Ú Nd4!94I=E+]?50!4B'*=E?G~aK<G?5N5Nd#0Q4+Q2%$8+ ZNa4IQ9I4=<[email protected]+Q254 }$&9 ?50!4Q025$U&4R0Q?(99?(6¦"a4IS+3`$i9I#6N5?(+34IU['*!?50V ÛMNd4!94=<+T&@+Q254PLì':=5S5 2T0 ?50!4Q0\>#4I++3254cUG':Q?50\$&=5S Ú Nd4Q9I4=<+&@+Q254 B':=[?[~b?(0Q4!0Z>#4+\+32(4PU['*!?50KDL4 0Q4I)*4I9I+T$¦Nd4!0Q#=$8+\3$&=(S5#6$&=5S)*4$&Q=7+32%$8+Z254Z0_-[0!+34I6 2\$&0Z'*[email protected]+Q4S 2T',+32 +32(4/U[':Q?50IKLì25$8+T'*0Z+Q254/N5Q#"5$&"5'*):':+.-+32%$¨+T0Q254'*0T$¦Lì'*=(S5 2T0c?(0Q4 K "d~9#=<+J$8'*=50 ¦9&'*=50T$&=(S|4$&9J2|2%$&0c$`S5S' Pa4!4=<+TN5Q&"%$&"5':)*':+.-&@]0Q2( 2T':=5> 254$&S50K B4I+ âJ²;å;å;åI² S54I=5&+34m+Q254mN5!#"%$&"('*)*',+.- &@/254;$8S50#=Ó4;$89J2Î9I#'*= K <G?5N(Na#0!4+Q2%$8+ aâ® Ú ² 5ãZ® "5²%ä\® "#Û[² ®rZÜ " $8=5 S \® å B4I+A± S54I=5&+34 &!2(4;$&S50':0#"(+J$8'*=54IS (/$&=5SÃ)*4+ \æ×S54=(&+34+32(4c4U#4I=E+A+325$8+9#':= í'*00!4)*4I9I+Q4S K Ïª$#Ò <G4):49I+i$9#':= $8+3$&=5S(#6 $&=5S +3&0Q0/':+RK <G?5N5Nd#0Q47$254;$&S '*0O&"(+J$&':=54S K Lì2%$8+]'*0+Q254Nd#0_+34!'*#N(Q#"%$8"5'*):':+.-H+Q2%$8+19&'*=í 2\$&010Q4):49+34SÏí1® ²å;å;å² #Ò $.=|&+3254I 2\#!S50V -5=5S \Ï \æ L ±mÒ@B#í]® ²;åå;å² [K Ï"%ÒP^]&0Q0R+32(4¦9#':=$&><$8'*= KHLì25$8+O'*0c+3254`N(Q#"%$8"5'*):':+.-&@A$&=(&+3254IR254$&S $.= &+Q254 2#QS50 -%=(S \Ïª±`ã L ±â¢Ò 2T254!4i± [ ® &!254;$8S50T#=|+Q#0Q40 ^%5K ( P 2Ó0!?5N5Nd#0Q4T+325$8++3254T4~GNd4!'*64=<+ 2\$&09;$8QQ':4S#?(+$&[email protected]#):)* 2T0K1L4T0Q4I)*4I9I+ $`9#':=|$8+T3$&=(S5#6 $&=(S|+Q#0Q0T':+T?(=E+Q'*)$`254$&S|'*0#"(+3$&'*=(4S K Ï9;Ò `x'*=5S \Ï \æ L + Ò 2T254!34 + ® &-%Q0_+P2(4;$&S|'*0#"G+J$&':=54S#=+3#0!0 5K)( Û Ú KÏ E[& P eK Ò <G?5N5Nd#0Q4 $ 9I#'*=Ó2%$80N5Q#"5$&"5'*):':+.- Î&@/@$&)*):'*=5> 254$&S50K $ @ 2\4OY%':N+3254O9I#'*=67$&=<-b+Q'*640IV 2K4 2#?5):S4~GNd49I+T+Q254ON5Q&Na#_+3':#= &@254$&S50\+Q¦"d4R=54;$8 KxL4 2T'*):)×6b'$ 84H+32('*0@B&Q67$&)×) $8+Q4Kx^x'$ 84 ® å Üi$&=5S ® ;Ú#Ú#Ú $8=5Sm0Q':6¦?5)*$8+34 9I#'*=|Y%'*N(0KR):&+P+Q254iN(Q#Nd#!+Q'*#=³&@254$&S50O$80R$ @B?5=59+3':#=&@ K #[email protected]#b® å Ú Ü(K Û KÏ <[# P eK %Ò <G?5N5Nd#0!N4 2\4Y%':N`$R9#':= +3':674I0D$&=(S):4I+ ÃS54=58+34 +32(47N5!#"%$&"('*)*',+.-}[email protected]$&S50K B4+ "d4Ã+Q254=[?(6¦"[email protected]$&S50K7L49;$&): ) ZM ( $ $ $ $ $ , !* , $ $ $ +7 P/ 9F 9F 1 5/ +7 P/ 9F 9F 1 5/ Ü * $ "5':=5#6' $&)TQ$&=5S5#6 U8$&Q'*$&"5)*4'2T25'*9J2ì':0S5'*0!9?50!0Q4ISX'*=¯+Q254=54e~G+9J25$&N(+34IK $.=<+3?(':+3':#=0!?5>#>#4I0!+Q0¦+32%$8+ 2T'*):)"d49):#0Q4+3 K}^]0!44':@\+325':0¦'*0+3Q?(4&V 24¦9;$8=Q4Nd4;$¨+R+325':0H4e~(Nd4!'*64=<+O67$&=<-³+Q'*640H$&=5S$;U#4I3$&>&4¦+3254 U8$&):?540IK Z$&!!-Ã#?G+Z$0Q'*6i?5) $8+Q'*#=Ã$8=5Sb9I#67N5$&Q4T+3254H$;U#4Q$&>#4P&@ +Q254 0+3 KD^]_+Q25'*[email protected]#®½å¿ÜÃ$&=5S ® Ú ² ;Ú#Ú ² Ú&Ú#Ú K Û#ÛGKMÏ HE[& H eK Ò P4IQ4 24 2T':)*)>#4+¦0Q&6744~GNd4Q':4=59I40Q':6¦?5)*$8+3':=5> 9I#=5S5',+3'*&=%$&)dN5Q#"5$&"5'*):':+Q'*40IK \&=50Q':S54+3&0Q0Q':=5>Ã$¦@$&'*S5'*48K B4I+ ¼ ®°8ÛG² 5² Gµ $&=(?S + ®° ²eÛ[²JÜ(² (µEK^254I= V \Ï ¼ Ò® "#Û[V \Ï +`Ò®XÛ "&Üc$&=5S \Ï ¼ +`Ò® "8Ü(K <G':=594 \Ï ¼ +`Ò® \Ï ¼ Ò \Ï +¦ÒV<+3254T4U#4=<+Q0 ¼ $&=5S + $&!4T'*=5S54INa4I=5S54I=E+4 K <G'*6i(? )*$8+34cS53$ [email protected]#6 +32(4H03$867N5):4P0!N%$&9I4O$&=5SbU&4!':@F-Ã+32%$¨+ Ï ¼ +`Ò® Ï ¼ Ò bÏ +`Ò 2T254IQ4 Ï ¼ Òi'*0¦+32(4N5Q#Nd#_+3'*&= &@c+3'*640 ¼ G9I9?5!Q4S '*= +Q2540Q':6¦?5)*$8+3':#= $&=(S70!'*6'*) $8Q):[email protected]& Ï ¼ +`Ò$&=5S 7Ï +`ÒeK P 2 -%=(S+ 2i4U#4=<+Q0 ¼ $&=5S ++Q2%$8+ $&!4R=5&+T':=5S54INa4I=5S54=<+K \&67N5?G+34 Ï ¼ Ò² 7Ï +`Ò$&=5S Ï ¼ +¦ÒK \#6N%$&Q4P+3254 9$&)*9I?5) $8+Q4SU8$&)*?(40O+3+3254I'*R+Q254#!4I+Q'*9;$8)DU&$8)*?54I0K #c4INa#_+/-##?5Q4I0Q?5),+30$&=5S ':=E+Q4!N5Q4+;K P/ 9F #7 9F 1 5/ * O$ $ $ 9$ $ ?$ * * * * * * * * * #! 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The basic question is this: what can we say about the limiting behavior of a sequence of random variables X1 , X2 , X3 , . . .? Since statistics and data mining are all about gathering data, we will naturally be interested in what happens as we gather more and more data. In calculus we say that a sequence of real numbers xn converges to a limit x if, for every ² > 0, |xn − x| < ² for all large n. In probability, convergence is more subtle. Going back to calculus for a moment, suppose that xn = x for all n. Then, trivially, limn xn = x. Consider a probabilistic version of this example. Suppose that X1 , X2 , . . . is a sequence of random variables which are independent and suppose each has a N (0, 1) distribution. Since these all have the same distribution, we are 90 6. Convergence of Random Variables tempted to say that Xn “converges” to X ∼ N (0, 1). But this can’t quite be right since P(Xn = Z) = 0 for all n. (Two continuous random variables are equal with probability zero.) Here is another example. Consider X1 , X2 , . . . where Xi ∼ N (0, 1/n). Intuitively, Xn is very concentrated around 0 for large n. But P (Xn = 0) = 0 for all n. This chapter develops appropriate methods of discussing convergence of random variables. There are two main ideas in this chapter: 1. The law of large numbers says that sample average P X n = n−1 ni=1 Xi converges in probability to the expectation µ = E(X). 2. The central limit theorem says that sample average has approximately a Normal distribution for large n. More √ precisely, n(X n − µ) converges in distribution to a Normal(0, σ 2 ) distribution, where σ 2 = V(X). 6.2 Types of Convergence The two main types of convergence are defined as follows. 6.2 Types of Convergence 91 Definition 6.1 Let X1 , X2 , . . . be a sequence of random variables and let X be another random variable. Let Fn denote the cdf of Xn and let F denote the cdf of X. P 1. Xn converges to X in probability, written Xn −→ X, if, for every ² > 0, P(|Xn − X| > ²) → 0 (6.1) as n → ∞. 2. Xn converges to X in distribution, written Xn Ã X, if, lim Fn (t) = F (t) (6.2) n→∞ at all t for which F is continuous. There is another type of convergence which we introduce mainly because it is useful for proving convergence in probability. Definition 6.2 Xn converges to X in quadratic mean qm (also called convergence in L2 ), written Xn −→ X, if, E(Xn − X)2 → 0 (6.3) as n → ∞. If X is a point mass at c – that is P(X = c) = 1 – we write qm qm P Xn −→ c instead of X −→ X. Similarly, we write Xn −→ c and Xn Ã c. Example 6.3 Let Xn ∼ N (0, 1/n). Intuitively, Xn is concentrating at 0 so we would like to say that Xn Ã 0. Let’s see if this is true. Let F be the distribution function for a point mass at 92 6. Convergence of Random Variables √ 0. Note that nXn ∼ N (0, 1). Let Z denote a standard normal √ random variable. For t < 0, Fn (t) = P(Xn < t) = P( nXn < √ √ √ nt) = P(Z < nt) → 0 since nt → −∞. For t > 0, √ √ √ Fn (t) = P(Xn < t) = P( nXn < nt) = P(Z < nt) → 1 √ since nt → ∞. Hence, Fn (t) → F (t) for all t 6= 0 and so Xn Ã 0. But notice that Fn (0) = 1/2 6= F (1/2) = 1 so convergence fails at t = 0. But that doesn’t matter because t = 0 is not a continuity point of F and the definition of convergence in distribution only requires convergence at continuity points. ¥ The next theorem gives the relationship between the types of convergence. The results are summarized in Figure 6.1. Theorem 6.4 The following relationships hold: qm P (a) Xn −→ X implies that Xn −→ X. P (b) Xn −→ X implies that Xn Ã X. (c) If Xn Ã X and if P(X = c) = 1 for some real number c, P then Xn −→ X. In general, none of the reverse implications hold except the special case in (c). qm Proof. We start by proving (a). Suppose that Xn −→ X. Fix ² > 0. Then, using Chebyshev’s inequality, E|Xn − X|2 → 0. P(|Xn − X| > ²) = P(|Xn − X| > ² ) ≤ ²2 Proof of (b). This proof is a little more complicated. You may skip if it you wish. Fix ² > 0 and let x be a continuity point of F . Then 2 2 Fn (x) = P(Xn ≤ x) = P(Xn ≤ x, X ≤ x + ²) + P(Xn ≤ x, X > x + ²) ≤ P(X ≤ x + ²) + P(|Xn − X| > ²) = F (x + ²) + P(|Xn − X| > ²). Also, F (x − ²) = P(X ≤ x − ²) = P(X ≤ x − ², Xn ≤ x) + P(X ≤ x + ², Xn > x) 6.2 Types of Convergence 93 ≤ Fn (x) + P(|Xn − X| > ²). Hence, F (x−²)−P(|Xn −X| > ²) ≤ Fn (x) ≤ F (x+²)+P(|Xn −X| > ²). Take the limit as n → ∞ to conclude that F (x − ²) ≤ lim inf Fn (x) ≤ lim sup Fn (x) ≤ F (x + ²). n→∞ n→∞ This holds for all ² > 0. Take the limit as ² → 0 and use the fact that F is continuous at x and conclude that limn Fn (x) = F (x). Proof of (c). Fix ² > 0. Then, P(|Xn − c| > ²) = P(Xn < c − ²) + P(Xn > c + ²) ≤ P(Xn ≤ c − ²) + P(Xn > c + ²) = Fn (c − ²) + 1 − Fn (c + ²) → F (c − ²) + 1 − F (c + ²) = 0 + 1 − 0 = 0. Let us now show that the reverse implications do not hold. Convergence in probability does not imply convergence in quadratic mean. Let U ∼ Unif(0, 1) and let √ √ Xn = nI(0,1/n) (U ). Then P(|Xn | > ²) = P( nI(0,1/n) (U ) > P ²) = P(0 ≤ U < 1/n) = 1/n → 0. Hence, Then Xn −→ 0. But R 1/n E(Xn2 ) = n 0 du = 1 for all n so Xn does not converge in quadratic mean. Convergence in distribution does not imply convergence in probability. Let X ∼ N (0, 1). Let Xn = −X for n = 1, 2, 3, . . .; hence Xn ∼ N (0, 1). Xn has the same distribution function as X for all n so, trivially, limn Fn (x) = F (x) d for all x. Therefore, Xn → X. But P(|Xn − X| > ²) = P(|2X| > ²) = P(|X| > ²/2) 6= 0. So Xn does not tend to X in probability. ¥ 94 6. Convergence of Random Variables point-mass distribution quadratic mean probability distribution FIGURE 6.1. Relationship between types of convergence. P We can conclude that E(Xn ) → b if Xn is uniformly integrable. See the technical appendix. Warning! One might conjecture that if Xn −→ b then E(Xn ) → b. This is not true. Let Xn be a random variable defined by P(Xn = n2 ) = 1/n and P(Xn = 0) = 1 − (1/n). Now, P(|Xn | < P ²) = P(Xn = 0) = 1 − (1/n) → 1. Hence, Z −→ 0. However, E(Xn ) = [n2 ×(1/n)]+[0×(1−(1/n))] = n. Thus, E(Xn ) → ∞. Summary. Stare at Figure 6.1. Some convergence properties are preserved under transformations. Theorem 6.5 Let Xn , X, Yn , Y be random variables. Let g be a continuous function. P P P (a) If Xn −→ X and Yn −→ Y , then Xn + Yn −→ X + Y . qm qm qm (b) If Xn −→ X and Yn −→ Y , then Xn + Yn −→ X + Y . (c) If Xn Ã X and Yn Ã c, then Xn + Yn Ã X + c. P P P (d) If Xn −→ X and Yn −→ Y , then Xn Yn −→ XY . (e) If Xn Ã X and Yn Ã c, then Xn Yn Ã cX. P P (f ) If Xn −→ X then g(Xn )−→ g(X). (g) If Xn Ã X then g(Xn ) Ã g(X). 6.3 The Law of Large Numbers Now we come to a crowning achievement in probability, the law of large numbers. This theorem says that the mean of a large sample is close to the mean of the distribution. For example, the proportion of heads of a large number of tosses is expected to be close to 1/2. We now make this more precise. 6.3 The Law of Large Numbers 95 Let X1 , X2 , . . . , be an iid sample and let µ = E(X1 ) and σ = V(X1 ). Recall that the sample mean is defined as X n = Note that µ = P E(Xi ) is the same n−1 ni=1 Xi and that E(X n ) = µ and V(X n ) = σ 2 /n. for all i so we can define µ = E(Xi ) for any i. By conTheorem 6.6 (The Weak Law of Large Numbers (WLLN).) P vention, we often If X1 , . . . , Xn are iid , then X n −→ µ. write µ = E(X1 ). 2 Interpretation of WLLN: The distribution of X n becomes more concentrated around µ as n gets large. Proof. Assume that σ < ∞. This is not necessary but it simplifies the proof. Using Chebyshev’s inequality, ¢ V(X n ) ¡ σ2 = P |X n − µ| > ² ≤ ²2 n²2 which tends to 0 as n → ∞. ¥ There is a stronger theorem in the appendix called the Example 6.7 Consider flipping a coin for which the probability strong law of large of heads is p. Let Xi denote the outcome of a single toss (0 numbers. or 1). Hence, p = P (Xi = 1) = E(Xi ). The fraction of heads after n tosses is X n . According to the law of large numbers, X n converges to p in probability. This does not mean that X n will numerically equal p. It means that, when n is large, the distribution of X n is tightly concentrated around p. Suppose that p = 1/2. How large should n be so that P (.4 ≤ X n ≤ .6) ≥ .7? First, E(X n ) = p = 1/2 and V(X n ) = σ 2 /n = p(1 − p)/n = 1/(4n). From Chebyshev’s inequality, P(.4 ≤ X n ≤ .6) = P(|X n − µ| ≤ .1) = 1 − P(|X n − µ| > .1) 1 25 ≥ 1− = 1 − . 4n(.1)2 n The last expression will be larger than .7 if n = 84. ¥ 96 6. Convergence of Random Variables 6.4 The Central Limit Theorem Suppose that X1 , . . . , Xn are iid with mean µ and variance σ. P The central limit theorem (CLT) says that X n = n−1 i Xi has a distribution which is approximately Normal with mean µ and variance σ 2 /n. This is remarkable since nothing is assumed about the distribution of Xi , except the existence of the mean and variance. Theorem 6.8 (The Central Limit Theorem (CLT).) Let X1 , . . . , Xn P be iid with mean µ and variance σ 2 . Let X n = n−1 ni=1 Xi . Then √ n(X n − µ) Zn ≡ ÃZ σ where Z ∼ N (0, 1). In other words, Z z 1 2 √ e−x /2 dx. lim P(Zn ≤ z) = Φ(z) = n→∞ 2π −∞ Interpretation: Probability statements about X n can be approximated using a Normal distribution. It’s the probability statements that we are approximating, not the random variable itself. In addition to Zn Ã N (0, 1), there are several forms of notation to denote the fact that the distribution of Zn is converging to a Normal. They all mean the same thing. Here they are: Zn ≈ N (0, 1) µ ¶ σ2 X n ≈ N µ, n µ ¶ σ2 X n − µ ≈ N 0, n ¡ ¢ √ n(X n − µ) ≈ N 0, σ 2 6.4 The Central Limit Theorem √ 97 n(X n − µ) ≈ N (0, 1). σ Example 6.9 Suppose that the number of errors per computer program has a Poisson distribution with mean 5. We get 125 programs. Let X1 , . . . , X125 be the number of errors in the programs. We want to approximate P(X < 5.5). Let µ = E(X1 ) = λ = 5 and σ 2 = V(X1 ) = λ = 5. Then, µ√ ¶ √ n(X n − µ) n(5.5 − µ) ≈ P(Z < 2.5) = .9938. < P(X < 5.5) = P σ σ √ The central limit theorem tells us that Zn = n(X − µ)/σ is approximately N(0,1). However, we rarely know σ. We can estimate σ 2 from X1 , . . . , Xn by n Sn2 = 1 X (Xi − X n )2 . n − 1 i=1 This raises the following question: if we replace σ with Sn is the central limit theorem still true? The answer is yes. Theorem 6.10 Assume the same conditions as the CLT. Then, √ n(X n − µ) Ã N (0, 1). Sn You might wonder, how accurate the normal approximation is. The answer is given in the Berry-Essèen theorem. Theorem 6.11 (Berry-Essèen.) Suppose that E|X1 |3 < ∞. Then sup |P(Zn ≤ z) − Φ(z)| ≤ z 33 E|X1 − µ|3 √ 3 . 4 nσ (6.4) There is also a multivariate version of the central limit theorem. 98 6. Convergence of Random Variables Theorem 6.12 (Multivariate central limit theorem) Let X1 , . . . , Xn be iid random vectors where X1i X2i Xi = .. . Xki with mean µ= µ1 µ2 .. . µk and variance matrix Σ. Let where X i = n−1 Pn = X= i=1 √ X1 X2 .. . Xk X1i . Then, E(X1i ) E(X2i ) .. . E(Xki ) . n(X − µ) Ã N (0, Σ). 6.5 The Delta Method If Yn has a limiting Normal distribution then the delta method allows us to find the limiting distribution of g(Yn ) where g is any smooth function. Theorem 6.13 (The Delta Method) Suppose that √ n(Yn − µ) Ã N (0, 1) σ and that g is a differentiable function such that g 0 (µ) 6= 0. Then √ n(g(Yn ) − g(µ)) Ã N (0, 1). |g 0 (µ)|σ 6.5 The Delta Method 99 In other words, ¶ µ ¶ µ 2 σ2 0 2σ Yn ≈ N µ, =⇒ g(Yn ) ≈ N g(µ), (g (µ)) . n n Example 6.14 Let X1 , . . . , Xn be iid with finite mean µ and fi√ nite variance σ 2 . By the central limit theorem, n(X n )/σ Ã N (0, 1). Let Wn = eX n . Thus, Wn = g(X n ) where g(s) = es . Since g 0 (s) = es , the delta method implies that Wn ≈ N (eµ , e2µ σ 2 /n). ¥ There is also a multivariate version of the delta method. Theorem 6.15 (The Multivariate Delta Method) Suppose that Yn = (Yn1 , . . . , Ynk ) is a sequence of random vectors such that √ n(Yn − µ) Ã N (0, Σ). Let g : Rk → R ane let ∂g ∂y1 ∇g(y) = ... . ∂g ∂yk Let ∇µ denote ∇g(y) evaluated at y = µ and assume that the elements of ∇µ are non-zero. Then √ ¢ ¡ n(g(Yn ) − g(µ)) Ã N 0, ∇Tµ Σ∇µ . Example 6.16 Let ¶ ¶ µ ¶ µ µ X1n X12 X11 , ..., , X2n X22 X21 be iid random vectors with mean µ = (µ1 , µ2 )T and variance Σ. Let n n 1X 1X X1 = X2 = X1i , X2i n i=1 n i=1 100 6. Convergence of Random Variables and define Yn = X 1 X 2 . Thus, Yn = g(X 1 , X 2 ) where g(s1 , s2 ) = s1 s2 . By the central limit theorem, µ ¶ √ X 1 − µ1 n Ã N (0, Σ). X 2 − µ2 Now ∇g(s) = Ã ! = ¶µ µ2 µ1 ∂g ∂s1 ∂g ∂s2 µ s2 s1 ¶ and so ∇Tµ Σ∇µ = (µ2 µ1 ) µ σ11 σ12 σ12 σ22 ¶ = µ22 σ11 + 2µ1 µ2 σ12 + µ21 σ22 . Therefore, √ ³ ´ 2 2 n(X 1 X 2 − µ1 µ2 ) Ã N 0, µ2 σ11 + 2µ1 µ2 σ12 + µ1 σ22 . ¥ 6.6 Bibliographic Remarks Convergence plays a central role in modern probability theory. For more details, see Grimmet and Stirzaker (1982), Karr (1993) and Billingsley (1979). Advanced convergence theory is explained in great detail in van der Vaart and Wellner (1996) and van der Vaart (1998). 6.7 Technical Appendix 6.7.1 Almost Sure and L1 Convergence We say that Xn converges almost surely to X, written as Xn −→ X, if P({s : Xn (s) → X(s)}) = 1. L 1 We say that Xn converges in L1 to X, written Xn −→ X, if E|Xn − X| → 0 6.7 Technical Appendix 101 as n → ∞. Theorem 6.17 Let Xn and X be random vaiables. Then: as P (a) Xn −→ X implies that Xn −→ X. qm L1 (b) Xn −→ X implies that Xn −→ X. L1 P (c) Xn −→ X implies that Xn −→ X. The weak law of large numbers says that X n converges to EX1 in probability. The strong law asserts that this is also true almost surely. Theorem 6.18 (The strong law of large numbers.) Let X1 , X2 , . . . as be iid. If µ = E|X1 | < ∞ then X n −→ µ. A sequence Xn is asymptotically uniformly integrable if lim lim sup E (|Xn |I(|Xn | > M )) = 0. M →∞ n→∞ P If Xn −→ b and Xn is asymptotically uniformly integrable, then E(Xn ) → b. 6.7.2 Proof of the Central Limit Theorem If X is a random variable, define its moment generating function (mgf) by ψX (t) = EetX . Assume in what follows that the mgf is finite in a neighborhood around t = 0. Lemma 6.19 Let Z1 , Z2 , . . . be a sequence of random variables. Let ψn the mgf of Zn . Let Z be another random variable and denote its mgf by ψ. If ψn (t) → ψ(t) for all t in some open interval around 0, then Zn Ã Z. proof of the central limit theorem. Let Yi = (Xi − P µ)/σ. Then, Zn = n−1/2 i Yi . Let ψ(t) be the mgf of Yi . The P √ mgf of i Yi is (ψ(t))n and mgf of Zn is [ψ(t/ n)]n ≡ ξn (t). 102 6. Convergence of Random Variables Now ψ 0 (0) = E(Y1 ) = 0, ψ 00 (0) = E(Y12 ) = V ar(Y1 ) = 1. So, ψ(t) = ψ(0) + tψ 0 (0) + t3 t2 00 ψ (0) + ψ 00 (0) + · · · 2! 3! t2 t3 00 + ψ (0) + · · · 2 3! t2 t3 00 = 1 + + ψ (0) + · · · 2 3! = 1+0+ Now, · µ ¶¸n t ξn (t) = ψ √ n ¸n · 2 t3 t 00 + ψ (0) + · · · = 1+ 2n 3!n3/2 #n " t3 t2 00 ψ (0) + · · · + 3!n1/2 = 1+ 2 n → et 2 /2 which is the mgf of a N(0,1). The result follows from the previous Theorem. In the last step we used the fact that, if an → a then ³ an ´n → ea . 1+ n 6.8 Excercises 1. Let X1 , . . . , Xn be iid with finite mean µ = E(X1 ) and finite variance σ 2 = V(X1 ). Let X n be the sample mean and let Sn2 be the sample variance. (a) Show that E(Sn2 ) = σ 2 . P P (b) Show that Sn2 −→ σ 2 . Hint: Show that Sn2 = cn n−1 ni=1 Xi2 − 2 dn X n where cn → 1 and dn → 1. Apply the law of large P numbers to n−1 ni=1 Xi2 and to X n . Then use part (e) of Theorem 6.5. 6.8 Excercises 103 2. Let X1 , X2 , . . . be a sequence of random variables. Show qm that Xn −→ b if and only if lim E(Xn ) = b and n→∞ lim V(Xn ) = 0. n→∞ 3. Let X1 , . . . , Xn be iid and let µ = E(X1 ). Suppose that qm the variance is finite. Show that X n −→ µ. 4. Let X1 , X2 , . . . be a sequence of random variables such that ¶ µ 1 1 1 = 1 − 2 and P (Xn = n) = 2 . P Xn = n n n Does Xn converge in probability? Does Xn converge in quadratic mean? 5. Let X1 , . . . , Xn ∼ Bernoulli(p). Prove that n 1X 2 P X −→ p and n i=1 i n 1 X 2 qm X −→ p. n i=1 i 6. Suppose that the height of men has mean 68 inches and standard deviation 4 inches. We draw 100 men at random. Find (approximately) the probability that the average height of men in our sample will be at least 68 inches. 7. Let λn = 1/n for n = 1, 2, . . .. Let Xn ∼ Poisson(λn ). P (a) Show that Xn −→ 0. P (b) Let Yn = nXn . Show that Yn −→ 0. 8. Suppose we have a computer program consisting of n = 100 pages of code. Let Xi be the number of errors on the ith page of code. Suppose that the Xi0 s are Poisson with mean P 1 and that they are independent. Let Y = ni=1 Xi be the total number of errors. Use the central limit theorem to approximate P(Y < 90). 104 6. Convergence of Random Variables 9. Suppose that P(X = 1) = P(X = −1) = 1/2. Define Xn = ½ X with probability 1 − en with probability n1 . 1 n Does Xn converge to X in probability? Does Xn converge to X in distribution? Does E(X − Xn )2 converge to 0? 10. Let Z ∼ N (0, 1). Let t > 0. (a) Show that, for any k > 0, P(|Z| > t) ≤ E|Z|k . tk (b) (Mill’s inequality.) Show that ½ ¾1/2 −t2 /2 2 e P(|Z| > t) ≤ . π t Hint. Note that P(|Z| > t) = 2P(Z > t). Now write out what P(Z > t) means and note that x/t > 1 whenever x > t. 11. Suppose that Xn ∼ N (0, 1/n) and let X be a random variable with distribution F (x) = 0 if x < 0 and F (x) = 1 if x ≥ 0. Does Xn converge to X in probability? (Prove or disprove). Does Xn converge to X in distribution? (Prove or disprove). 12. Let X, X1 , X2 , X3 , . . . be random variables that are positive and integer valued. Show that Xn Ã X if and only if lim P(Xn = k) = P(X = k) n→∞ for every integer k. 6.8 Excercises 105 13. Let Z1 , Z2 , . . . be i.i.d., random variables with density f . Suppose that P(Zi > 0) = 1 and that λ = limx↓0 f (x) > 0. Let Xn = n min{Z1 , . . . , Zn }. Show that Xn Ã Z where Z has an exponential distribution with mean 1/λ. 2 14. Let X1 , . . . , Xn ∼ Uniform(0, 1). Let Yn = X n . Find the limiting distribution of Yn . 15. Let µ X11 X21 ¶ µ ¶ µ ¶ X12 X1n , , ..., X22 X2n be iid random vectors with mean µ = (µ1 , µ2 ) and variance Σ. Let n n 1X 1X X1 = X1i , X 2 = X2i n i=1 n i=1 and define Yn = X 1 /X 2 . Find the limiting distribution of Yn . Part II Statistical Inference 103 7 Models, Statistical Inference and Learning 7.1 Introduction Statistical inference, or \learning" as it is called in computer science, is the process of using data to infer the distribution that generated the data. The basic statistical inference problem is this: We observe X1 ; : : : ; Xn F . We want to infer (o r estimate o r learn) F o r some feature of F such as its mean. 7.2 Parametric and Nonparametric Models A statistical model is a set of distributions (or a set of densities) F. A parametric model is a set F that can be parameterized b y a nite n umber of parameters. For example, if we assume that the data come from a Normal distribution then 106 7. Models, Statistical Inference and Learning the model is F = f (x; ; ) = 1 p exp 1 (x 2 2 )2 ; 2 R ; > 0 : (7.1) This is a two-parameter model. We have written the density as f (x; ; ) to show that x is a value of the random variable whereas and are parameters. In general, a parametric model takes the form F = f (x; ) : 2 (7.2) where is an unknown parameter (or vector of parameters) that can take values in the parameter space . If is a vector but we are only interested in one component of , we call the remaining parameters nuisance parameters. A nonparametric model is a set F that cannot be parameterized by a nite number of parameters. For example, FALL = fall cdf 0 sg is nondistinction parametric. The between parametric and nonparametric is more subtle than this but we don't need a rigorous denition for our purposes. Example 7.1 (One-dimensional Parametric Estimation.) Let X1 ; : : : ; Xn be independent Bernoulli(p) observations. The problem is to estimate the parameter p. Example 7.2 (Two-dimensional Parametric Estimation.) Suppose that X1 ; : : : ; Xn F and we assume that the pdf f 2 F where F is given in (7.1). In this case there are two parameters, and . The goal is to estimate the parameters from the data. If we are only interested in estimating then is the parameter of interest and is a nuisance parameter. Example 7.3 (Nonparametric estimation of the cdf.) Let X1 ; : : : ; Xn be independent observations from a cdf F . The problem is to estimate F assuming only that F 2 FALL = fall cdf 0 sg. Example 7.4 (Nonparametric density estimation.) Let X1 ; : : : ; Xn be independent observations from a cdf F and let f = F 0 be the 7.2 Parametric and Nonparametric Models 107 . Suppose we want to estimate the pdf f . It is not possible to estimate f assuming only that F 2 FALL . We need to assume some smoothness on f . For example, we might assume T that f 2 F = FDENS FSOB where FDENS is the set of all probability density functions and pdf FSOB = f : Z (f 00 (x))2 dx < 1 : The class FSOB is called a Sobolev space; it is the set of functions that are not \too wiggly." Example 7.5 (Nonparametric estimation of functionals.) Let X1 ; : : : ; Xn R F . Suppose we want to estimate = E (X1 ) = x dF (x) assuming only that exists. The mean may be thought of as a R function of F : we can write = T (F ) = x dF (x). In general, any function of F is called a statistical functional . Other R 2 examples of functions are the variance T (F ) = x dF (x) R 2 xdF (x) and the median T (F ) = F 1=2 . Example 7.6 (Regression, prediction and classication.) Suppose we observe pairs of data (X1 ; Y1 ); : : : (Xn ; Yn). Perhaps Xi is the blood pressure of subject i and Yi is how long they live. X is called a predictor or regressor or feature or independent variable. Y is called the outcome or the response variable or the dependent variable. We call r(x) = E (Y jX = x) the regression function. If we assume that f 2 F where F is nite dimensional { the set of straight lines for example { then we have a parametric regression model. If we assume that f 2 F where F is not nite dimensional then we have a parametric regression model. The goal of predicting Y for a new patient based on their X value is called prediction. If Y is discrete (for example, live or die) then prediction is instead called classication. If our goal is to estimate the functin f , then we call this regression or curve estimation. Regression models 108 7. Models, Statistical Inference and Learning are sometimes written as Y = f (X ) + (7.3) where E () = 0. We can always rewrite a regression model this way. To see this, dene = Y f (X ) and hence Y = Y + f (X ) f (X ) = f (X )+. Moreover, E () = E E (jX ) = E (E (Y f (X ))jX ) = E (E (Y jX ) f (X )) = E (f (X ) f (X )) = 0. It is traditional in most introductory courses to start with parametric inference. Instead, we will start with nonparametric inference and then we will cover parametric inference. In some respects, nonparametric inference is easier to understand and is more useful than parametric inference. What's Next? There are many approaches to statistical inference. The two dominant approaches are called frequentist inference and Bayesian inference. We'll cover both but we will start with frequentist inference. We'll postpone a discussion of the pro's and con's of these two until later. Frequentists and Bayesians. If F = ff (xR; ) : 2 g is a parametric model, we write P (X 2 A) = A f (x; )dx and E (r(X )) = R r(x)f (x; )dx. The subscript indicates that the probability or expectation is with respect to f (x; ); it does not mean we are averaging over . Similarly, we write V for the variance. Some Notation. 7.3 Fundamental Concepts in Inference Many inferential problems can be identied as being one of three types: estimation, condence sets or hypothesis testing. We will treat all of these problems in detail in the rest of the book. Here, we give a brief introduction to the ideas. 7.3.1 Point Estimation 7.3 Fundamental Concepts in Inference 109 Point estimation refers to providing a single \best guess" of some quantity of interest. The quantity of interest could be a parameter in a parametric model, a cdf F , a probability density function f , a regression function r, or a prediction for a future value Y of some random variable. By convention, we denote a point estimate of by b. Remember that is a xed, unknown quantity. The estimate b depends on the data so b is a random variable. More formally, let X1 ; : : : ; Xn be n iid data point from some distribution F . A point estimator bn of a parameter is some function of X1 ; : : : ; Xn: bn = g (X1; : : : ; Xn ): We dene bias (bn ) = E (bn ) (7.4) to be the bias of bn . We say that bn is unbiased if E (bn ) = . Unbiasedness used to receive much attention but these days it is not considered very important; many of the estimators we will use are biased. A point estimator bn of a parameter is consistent if bn P! . Consistency is a reasonable requirement for estimators. The distribution of bn is called the sampling distribution. The standard deviation of bn is called the standard error, denoted by se : se = se (bn ) q = V (bn ): (7.5) Often, it is not possible to compute the standard error but usually we can estimate the standard error. The estimated standard error is denoted by seb . Example 7.7 Let X1 ; : : : ; Xn Bernoulli(p) and let pbn = n 1 P Xi . i P Then E (pbn ) = n 1 i E (Xi ) = p so pbn is unbiased. The standard 110 7. Models, Statistical Inference and Learning p p error is se = p V (pbn ) = p(1 error is seb = pb(1 pb)=n. p)=n. The estimated standard The quality of a point estimate is sometimes assessed by the mean squared error, or MSE, dened by MSE = E (bn )2 : Recall that E () refers to expectation with respect to the distribution n Y f (x1 ; : : : ; xn ; ) = f (xi ; ) i=1 that generated the data. It does not mean we are averaging over a density for . Theorem 7.8 The MSE can be written as MSE = bias (bn )2 + V (bn ): Proof. E (bn (7.6) Let n = E (bn ). Then )2 = E (bn n + n )2 = E (bn n)2 + 2(n )2 E (bn = (n )2 + E (bn n )2 = bias 2 + V (bn ): n ) + E ( n !P 0 and se ! 0 as n ! 1 then bn is consistent, that is, bn ! . Proof. If bias ! 0 and se ! 0 then, by Theorem 7.8, qm MSE ! 0. It follows that bn ! . (Recall denition 6.3.) The result follows from part (b) of Theorem 6.4. Theorem 7.9 If bias Example 7.10 Returning to the coin ipping example, we have p that E p (pbn ) = p so that bias = p p = 0 and se = p(1 p)=n ! 0. Hence, pbn P! p, that is, pbn is a consistent estimator. )2 7.3 Fundamental Concepts in Inference 111 Many of the estimators we will encounter will turn out to have, approximately, a Normal distribution. Denition 7.11 An estimator is asymptotically Nor- mal if bn se N (0; 1): (7.7) 7.3.2 Condence Sets A 1 condence interval for a parameter is an interval Cn = (a; b) where a = a(X1 ; : : : ; Xn ) and b = b(X1 ; : : : ; Xn ) are functions of the data such that P ( 2 Cn) 1 ; for all 2 : (7.8) In words, (a; b) traps with probability 1 . We call 1 the coverage of the condence interval. Commonly, people use 95 per cent condence intervals which corresponds to choosing = 0:05. Note: Cn is random and is xed! If is a vector then we use a condence set (such a sphere or an ellipse) instead of an interval. Warning! There is much confusion about how to interpret a condence interval. A condence interval is not a probability statement about since is a xed quantity, not a random variable. Some texts interpret condence intervals as follows: if I repeat the experiment over and over, the interval will contain the parameter 95 per cent of the time. This is correct but useless since we rarely repeat the same experiment over and over. A better interpretation is this: On day 1, you collect data and construct a 95 per cent condence interval for a parameter 1 . On day 112 7. Models, Statistical Inference and Learning 2, you collect new data and construct a 95 per cent condence interval for an unrelated parameter 2 . On day 3, you collect new data and construct a 95 per cent condence interval for an unrelated parameter 3 . You continue this way constructing condence intervals for a sequence of unrelated parameters 1 ; 2 ; : : : Then 95 per cent your intervals will trap the true parameter value. There is no need to introduce the idea of repeating the same experiment over and over. Example 7.12 Every day, newspapers report opinion polls. For example, they might say that \83 per cent of the population favor arming pilots with guns." Usually, you will see a statement like \this poll is accurate to within 4 points 95 per cent of the time." They are saying that 83 4 is a 95 per cent condence interval for the true but unknown proportion p of people who favor arming pilots with guns. If you form a condence interval this way everyday for the rest of your life, 95 per cent of your intervals will contain the true parameter. This is true even though you are estimating a dierent quantity (a dierent poll question) every day. Later, we will discuss Bayesian methods in which we treat as if it were a random variable and we do make probability statements about . In particular, we will make statements like \the probability that is Cn, given the data, is 95 per cent." However, these Bayesian intervals refer to degree-of-belief probabilities. These Bayesian intervals will not, in general, trap the parameter 95 per cent of the time. Example 7.13 In the coin ipping setting, let Cn = (pbn n ; pbn + n ) where 2n = log(2=)=(2n). From Hoeding's inequality (5.4) it follows that P(p 2 Cn ) 1 for every p. Hence, Cn is a 1 condence interval. 7.3 Fundamental Concepts in Inference 113 As mentioned earlier, point estimators often have a limiting Normal distribution, meaning that equation (7.7) holds, that is, bn N (; seb 2 ). In this case we can construct (approximate) condence intervals as follows. Theorem 7.14 (Normal-based Condence Interval.) Suppose that bn N (; seb 2 ). Let be the cdf of a standard Normal and let z=2 = 1 (1 (=2)), that is, P(Z > z=2 ) = =2 and P( z=2 < Z < z=2 ) = 1 where Z N (0; 1). Let Cn = (bn Then z=2 seb ; bn + z=2 seb ): P ( 2 Cn) ! 1 : (7.10) Let Zn = (bn )=seb . By assumption Zn Z N (0; 1). Hence, Proof. P ( 2 Cn) (7.9) = P bn = P ! P z=2 seb < z=2 < bn b se z=2 < Z < z=2 = 1 : Z where < bn + z=2 seb ! < z=2 For 95 per cent condence intervals, = 0:05 and z=2 = 1:96 2 leading to the approximate 95 per cent condence interval bn 2seb . We will discuss the construction of condence intervals in more generality in the rest of the book. Example 7.15 Let X1 ; : : : ; Xn P Bernoulli( p) and let pbn = n P 1 Pn Xi . i=1 2 n p(1 p) = n 2 np(1 Then V (pbn ) = n 2 ni=1 V (Xp i) = n i=1 p p) = p(1 p)=n. Hence, se = p(1 p)=n and seb = pbn (1 pbn )=n. By the Central Limit Theorem, pbn N (p; se b 2 ). Therefore, an approximate 1 condence interval is pb z=2 seb = pb z=2 r pbn (1 pbn ) : n 114 7. Models, Statistical Inference and Learning Compare this with the condence interval in the previous example. The Normal-based interval is shorter but it only has approximately (large sample) correct coverage. 7.3.3 Hypothesis Testing The term \retaining the null hypothesis" is due to Chris Genovese. Other terminology is \accepting the null" or \failing to reject the null." In hypothesis testing, we start with some default theory { called a null hypothesis { and we ask if the data provide suÆcient evidence to reject the theory. If not we retain the null hypothesis. Example 7.16 (Testing if a Coin is Fair) Suppose X1 ; : : : ; Xn Bernoulli(p) denote n independent coin ips. Suppose we want to test if the coin is fair. Let H0 denote the hypothesis that the coin is fair and let H1 denote the hypothesis that the coin is not fair. H0 is called the null hypothesis and H1 is called the alternative hypothesis. We can write the hypotheses as H0 : p = 1=2 versus H1 : p 6= 1=2: It seems reasonable to reject H0 if T = jpbn (1=2)j is large. When we discuss hypothesis testing in detail, we will be more precise about how large T should be to reject H0 . 7.4 Bibliographic Remarks Statistical inference is covered in many texts. Elementary texts include DeGroot and Schervish (2001) and Larsen and Marx (1986). At the intermediate level I recommend Casella and Berger (2002) and Bickel and Doksum (2001). At the advanced level, Lehmann and Casella (1998), Lehmann (1986) and van der Vaart (1998). 7.5 Technical Appendix 7.5 Technical Appendix 115 Our denition of condence interval requires that P ( 2 Cn ) 1 for all 2 . An pointwise asymptotic condence interval requires that lim infn!1 P ( 2 Cn ) 1 for all 2 . An uniform asymptotic condence interval requres that lim infn!1 inf 2 P ( 2 Cn ) 1 . The approximate Normal-based interval is a pointwise asymptotic condence interval. In general, it might not be a uniform asymptotic condence interval. 116 7. Models, Statistical Inference and Learning 8 Estimating the cdf and Statistical Functionals The rst inference problem we will consider is nonparametric estimationthe of cdf F and functions of the cdf. 8.1 The Empirical Distribution Function Let X1 ; : : : ; Xn F be iid where F is a distribution function on the real line. We will estimate F with the empirical distribution function, which is dened as follo ws. Denition 8.1 The empirical distribution function Fbn is the cdf that puts mass 1=n at e ach data point Xi . Formally, Fbn (x) = Pn i=1 I (Xi x) n n umber of observations less than or equal to x = (8.1) n where I (Xi x) = 1 if Xi x 0 if Xi > x: 8. Estimating the cdf and Statistical Functionals 118 Example 8.2 (Nerve Data.) Cox and Lewis (1966) reported 799 waiting times between successive pulses along a nerve bre. The rst plot in Figure 8.1 shows the a \toothpick plot" where each toothpick shows the location of one data point. The second plot shows that empirical cdf Fbn . Suppose we want to estimate the fraction of waiting times between .4 and .6 seconds. The estimate is Fbn (:6) Fbn (:4) = :93 :84 = :09. The following theorems give some properties of Fbn (x). Theorem 8.3 At any xed value of x, E Fbn (x) = F (x) and V Fbn (x) Thus, F (x)(1 F (x)) n P and hence, Fbn (x) ! F (x). MSE = = F (x)(1 F (x)) : n !0 Actually, supx jFbn (x) F (x)j Theorem 8.4 (Glivenko-Cantelli Theorem.) Let X1 ; : : : ; Xn F . converges to 0 Then almost surely. sup jFbn(x) F (x)j P! 0: x 8.2 Statistical Functionals A statistical functional T (F ) is any function Rof F . Examples R are the mean = xdF (x), the variance 2 = (x )2 dF (x) and the median m = F 1 (1=2). Denition 8.5 The plug-in estimator of = T (F ) is dened by bn = T (Fbn ): In other words, just plug in Fbn for the unknown F . 8.2 Statistical Functionals 0.0 0.2 0.4 0.6 119 0.8 1.0 1.2 1.4 0.8 1.0 1.2 1.4 0.4 0.0 cdf 0.8 time 0.0 0.2 0.4 0.6 time FIGURE 8.1. Nerve data. The solid line in the middle is the empirical distribution function. The lines above and below the middle line are a 95 per cent condence band. The condence band is explained in the appendix. 120 8. Estimating the cdf and Statistical Functionals R A functional of theR form r(x)dF (x) is called a Rlinear functional. Recall that r(x)dFP(x) is dened to be r(x)f (x)dx in the continuous case and j r(xj )f (xj ) in the discrete. The empirical cdf Fbn (x)R is discrete, putting mass 1=n at each Xi . Hence, if T (F ) = r(x)dF (x) is a linear functional then we have: R The plug-in estimator for linear functional T (F ) = r(x)dF (x) is: T (Fbn) = Z r(x)dFbn (x) n 1X = r(Xi ): n i=1 (8.2) Sometimes we can nd the estimated standard error se b of T (Fbn ) by doing some calculations. However, in other cases it is not obvious how to estimate the standard error. In the next chapter, we will discuss a general method for nding se b . For now, let us just assume that somehow we can nd se b . In many cases, it turns out that 2 T (Fbn ) N (T (F ); se b ): By equation (7.10), an approximate 1 for T (F ) is then condence interval T (Fbn ) z=2 se: b We will call this the Normal-based interval. For a 95 per cent condence interval, z=2 = z:05=2 = 1:96 2 so the interval is T (Fbn ) 2 se: b Example 8.6 (The mean.) Let = T (F ) = R R x dF (x). The plugin estimator is b = x dFbn (x) = X n . The standard error is q p se = V (X n ) = = n. If b denotes an estimate of , then (8.3) 8.2 Statistical Functionals p 121 the estimated standard error is b= n. (In the next example, we shall see how to estimate .) A Normal-based condence interval for is X n z=2 se2 : Example 8.7 (The Variance) Let 2 = T (F ) = V (X ) = R xdF (x) 2 . The plug-in estimator is b2 = = Z R 2 x dF (x) Z 2 b xdFn (x) !2 n X x2 dFbn(x) n 1X X2 n i=1 i 1 X n i=1 i n 1X = (X n i=1 i X n )2 : Another reasoable estimator of 2 is the sample variance Sn2 = n X 1 (Xi X n )2 : n 1 i=1 In practice, there is little dierence between b2 and Sn2 and you can use either one. Returning the our last example, we now see that the estimated standard error of the estimate of the mean is p se b = b = n. Example 8.8 (The Skewness) Let and 2 denote the mean and variance of a random variable X . The skewness is dened to be R E (X )3 = = 3 R (x )3 dF (x) : (x )2 dF (x) 3=2 The skewness measure the lack of symmetry of a distribution. P To nd the plug-in estimate, rst recall that b = n 1 i Xi and P b2 = n 1 i (Xi b)2 . The plug-in estimate of is b = R nR (x (x 1 P (Xi n i o3=2 = b3 )3 dFbn(x) )2 dFbn(x) b)3 : 122 8. Estimating the cdf and Statistical Functionals Example 8.9 (Correlation.) Let Z = (X; Y ) and let = T (F ) = E (X X )(Y Y )=(x y ) denote the correlation between X and Y , where F (x; y ) is bivariate. We can write T (F ) = a(T1 (F ); T2 (F ); T3 (F ); T4 (F ); T5 (F )) where R R R T1 (F ) = R x dF (z ) T2 (F ) = R y dF (z ) T3 (F ) = xy dF (z ) T4 (F ) = x2 dF (z ) T5 (F ) = y 2 dF (z ) and a(t1 ; : : : ; t5 ) = p (t4 t3 t1 t2 : t21 )(t5 t22 ) Replace F with Fbn in T1 (F ), . . . , T5 (F ), and take b = a(T1 (Fbn ); T2 (Fbn ); T3 (Fbn ); T4 (Fbn ); T5 (Fbn )): We get b = qP P i (Xi i (Xi X n )(Yi Y n ) qP 2 X n )2 i (Yi Y n ) which is called the sample correlation. Example 8.10 (Quantiles.) Let F be strictly increasing with den- sity f . The T (F ) = F 1 (p) be the pth quantile. The estimate if T (F ) is Fbn 1 (p). We have to be a bit careful since Fbn is not invertible. To avoid ambiguity we dene Fbn 1 (p) = inf fx : Fbn (x) pg. We call Fbn 1 (p) the pth sample quantile. Only in the rst example did we compute a standard error or a condence interval. How shall we handle the other examples. When we discuss parametric methods, we will develop formulae for standard errors and condence intervals. But in our nonparametric setting we need something else. In the next chapter, we will introduce two methods { the jackknife and the bootstrap { for getting standard errors and condence intervals. Example 8.11 (Plasma Cholesterol.) Figure 8.2 shows histograms for plasma cholesterol (in mg/dl) for 371 patients with chest 8.2 Statistical Functionals 123 pain (Scott et al. 1978). The histograms show the percentage of patients in 10 bins. The rst histogram is for 51 patients who had no evidence of heart disease while the second histogram is for 320 patients who had narrowing of the arteries. Is the mean cholesterol dierent in the two groups? Let us regard these data R as samples Rfrom two distributions F1 and F2 . Let 1 = xdF1 (x) and 2 = xdF2 (x) denote the means of the two populations. R The plug-in estimates are b1 = xdFbn;1 (x) = X n;1 = 195:27 R and b2 = xdFbn;2 (x) = X n;2 = 216 :19: Recall that the standard Pn 1 error of the sample mean b = n i=1 Xi is se ( b) = v u n X u tV 1 Xi n i=1 = v u u t n 1 X V (Xi ) = n2 i=1 r n 2 =p 2 n n which we estimate by b ( se b) = where pbn v u n u1 X t b = (X n i=1 i X )2 : For the two groups this yields seb (b1 ) = 5:0 and seb (b2 ) = 2:4. Approximate 95 per cent condence intervals for 1 and 2 are b1 2seb (b1 ) = (185; 205) and b2 2seb (b2 ) = (211; 221). Now, consider the functional = T (F2 ) T (F1 ) whose plug-in estimate is b = b2 b1 = 216:19 195:27 = 20:92: The standard error of b is se = p V ( b2 b1 ) = p V ( b2 ) + V ( b1 ) = p (se (b1 ))2 + (se (b2 ))2 and we estimate this by b = se p (seb (b1 ))2 + (seb (b2 ))2 = 5:55: 124 8. Estimating the cdf and Statistical Functionals An approximate 95 per cent condence interval for is b 2seb = (9:8; 32:0). This suggests that cholesterol is higher among those with narrowed arteries. We should not jump to the conclusion (from these data) that cholesterol causes heart disease. The leap from statistical evidence to causation is very subtle and is discussed later in this text. 8.3 Technical Appendix In this appendix we explain how to construct a condence band for the cdf . Theorem 8.12 (Dvoretzky-Kiefer-Wolfowitz (DKW) inequality.) Let X1 ; : : : ; Xn be iid from F . Then, for any > 0, P sup jF (x) x j > 2e Fbn (x) 2n2 : (8.4) From the DKW inequality, we can construct a condence set. Let 2n = log(2=)=(2n), L(x) = maxfFbn (x) n ; 0g and U (x) = minfFbn (x) + n ; 1g. It follows from (8.4) that for any F , P (F 2 Cn) 1 : Thus, Cn is a nonparametric 1 condence set for F . A better name for Cn is a condence band. To summarize: A 1 nonparametric condence band for F is (L(x); U (x)) where L(x) = maxfFbn (x) n ; 0g U (x) = minfFbn (x) + n ; 1g s 1 2 n = log : 2n 8.3 Technical Appendix 100 150 200 250 300 125 350 400 plasma cholesterol for patients without heart disease 100 150 200 250 300 350 400 plasma cholesterol for patients with heart disease FIGURE 8.2. Plasma cholesterol for 51 patients with no heart disease and 320 patients with narrowing of the arteries. 126 8. Estimating the cdf and Statistical Functionals Example 8.13 The dashed lines 8.1 give a 95 per cent q in Figure condence band using n = 2 1 2n log :05 = :048. 8.4 Bibliographic Remarks The Glivenko-Cantelli theorem is the tip of the iceberg. The theory of distribution functions is a special case of what are called empirical processes which underlie much of modern statistical theory. Some references on empirical processes are Shorack and Wellner (1986) and van der Vaart and Wellner (1996). 8.5 Exercises 1. Prove Theorem 8.3. 2. Let X1 ; : : : ; Xn Bernoulli(p) and let Y1 ; : : : ; Ym Bernoulli(q ). Find the plug-in estimator and estimated standard error for p. Find an approximate 90 per cent condence interval for p. Find the plug-in estimator and estimated standard error for p q . Find an approximate 90 per cent condence interval for p q . 3. (Computer Experiment.) Generate 100 observations from a N(0,1) distribution. Compute a 95 per cent condence band for the cdf F . Repeat this 1000 times and see how often the condence band contains the true distribution function. Repeat using data from a Cauchy distribution. 4. Let X1 ; : : : ; Xn F and let Fbn (x) be the empirical distribution function. For a xed x, use the central limit theorem to nd the limiting distribution of Fbn (x). 5. Let x and y be two distinct points. Find Cov(Fbn (x); Fbn (y )). 8.5 Exercises 127 6. Let X1 ; : : : ; Xn F and let Fb be the empirical distribution function. Let a < b be xed numbers and dene = T (F ) = F (b) F (a). Let b = T (Fbn) = Fbn (b) Fbn (a). Find the estimated standard error of b. Find an expression for an approximate 1 condence interval for . 7. Data on the magnitudes of earthquakes near Fiji are available on the course website. Estimate the cdf F (x). Compute and plot a 95 per cent condence envelope for F . Find an approximate 95 per cent condence interval for F (4:9) F (4:3). 8. Get the data on eruption times and waiting times between eruptions of the old faithful geyser from the course website. Estimate the mean waiting time and give a standard error for the estimate. Also, give a 90 per cent condence interval for the mean waiting time. Now estimate the median waiting time. In the next chapter we will see how to get the standard error for the median. 9. 100 people are given a standard antibiotic to treat an infection and another 100 are given a new antibiotic. In the rst group, 90 people recover; in the second group, 85 people recover. Let p1 be the probability of recovery under the standard treatment and let p2 be the probability of recovery under the new treatment. We are interested in estimating = p1 p2 . Provide an estimate, standard error, an 80 per cent condence interval and a 95 per cent condence interval for . 10. In 1975, an experiment was conducted to see if cloud seeding produced rainfall. 26 clouds were seeded with silver nitrate and 26 were not. The decision to seed or not was made at random. Get the data from http://lib.stat.cmu.edu/DASL/Stories/CloudSeeding.html 128 8. Estimating the cdf and Statistical Functionals Let be the dierence in the median precipitation from the two groups. Estimate . Estimate the standard error of the estimate and produce a 95 per cent condence interval. 9 The Bootstrap The bootstrap is a nonparametric method for estimating standard errors and computing condence intervals. Let Tn = g (X1 ; : : : ; Xn) be a statistic, that is, any function of the data. Suppose we want to know V F (Tn ), the v ariance of Tn . We hav e written V F to emphasize that the variance usually depends onP the unknown n 1 distribution function F . ForRexample, if Tn = n X then i=1 R i 2 2 2 V F (Tn ) = =n where = (x ) dF (x) and = xdF (x). The bootstrap idea has two parts. First we estimate V F (Tn ) with Pn 1 V Fbn (Tn ). Thus, for Tn = n b 2 =n i=1 Xi we hav e V Fbn (Tn ) = P complicated where b2 = n 1 ni=1 (Xi X n ). However, in more cases we cannot write down a simple formula for V Fbn (Tn ). This leads us to the second step which is to approximate V Fbn (Tn ) using simulation. Before proceeding, let us discuss the idea of simulation. 9.1 Simulation 130 9. The Bootstrap Suppose we draw an iid sample Y1 ; : : : ; YB from a distribution G. By the law of large numbers, B 1X Yj P! Yn = B j =1 Z y dG(y ) = E (Y ) as B ! 1. So if we draw a large sample from G, we can use the sample mean Y n to approximate E (Y ). In a simulation, we can make B as large as we like in which case, the dierence between Y n and E (Y ) is negligible. More generally, if h is any function with nite mean then B 1X h(Y ) P! B j =1 j Z h(y )dG(y ) = E (h(Y )) as B ! 1. In particular, 2 Z 2 Z B B B 1X 1X 1X P 2 2 2 (Y Y ) = Y Y ! y dF (y) ydF (y ) = V (Y ): B j =1 j B j =1 j B j =1 j Hence, we can use the sample variance of the simulated values to approximate V (Y ). 9.2 Bootstrap Variance Estimation According to what we just learned, we can approximate V Fbn (Tn ) by simulation. Now V Fbn (Tn ) means \the variance of Tn if the distribution of the data is Fbn ." How can we simulate from the distribution of Tn when the data are assumed to have distribution Fbn ? The answer is to simulate X1 ; : : : ; Xn from Fbn and then compute Tn = g (X1; : : : ; Xn ). This constitutes one draw from the distribution of Tn . The idea is illustrated in the following diagram: Real world F =) X1 ; : : : ; Xn =) Tn = g (X1; : : : ; Xn ) Bootstrap world Fbn =) X1 ; : : : ; Xn =) Tn = g (X1; : : : ; Xn ) 9.2 Bootstrap Variance Estimation 131 How do we simulate X1 ; : : : ; Xn from Fbn ? Notice that Fbn puts mass 1/n at each data point X1 ; : : : ; Xn. Therefore, drawing an observation from Fbn is equivalent to drawing one point at random from the original data set. Thu, to simulate X1 ; : : : ; Xn Fbn it suÆces to draw n observations with replacement from X1 ; : : : ; Xn . Here is a summary: Boostrap Variance Estimation 1. Draw X1 ; : : : ; Xn Fbn . 2. Compute Tn = g (X1; : : : ; Xn ). . 3. Repeat steps 1 and 2, B times, to get Tn; 1 ; : : : ; Tn;B 4. Let B 1X vboot = T B b=1 n;b B 1X T B r=1 n;b !2 : (9.1) Example 9.1 The following pseudo-code shows how to use the bootstrap to estimate the standard of the median. 132 9. The Bootstrap Bootstrap for The Median Given data X = (X(1), ..., X(n)): T <- median(X) Tboot <- vector of length B for(i in 1:N){ Xstar <- sample of size n from X (with replacement) Tboot[i] <- median(Xstar) } se <- sqrt(variance(Tboot)) The following schematic diagram will remind you that we are using two approximations: V F (Tn ) not so small z}|{ V Fbn (Tn ) small z}|{ vboot : Example 9.2 Consider the nerve data. Let = T (F ) = R (x )3 dF (x)= 3 be the skewness. The skewness is a measure of asymmetry. A Normal distribution, for example, has skewness 0. The plug-in estimate of the skewness is b = T (Fbn ) = R (x P )3 dFbn (x) n1 ni=1 (Xi = b3 b3 X n )3 = 1:76: To estimate the standard error with the bootstrap we follow the same steps as with the medin example except we compute the skewness from each bootstrap sample. When applied to the nerve data, the bootstrap, based on B = 1000 replications, yields a standard error for the estimated skewness of .16. 9.3 Bootstrap Condence Intervals 133 9.3 Bootstrap Condence Intervals There are several ways to construct bootstrap condence intervals. Here we discuss three methods. Normal Interval. The simplest is the Normal interval Tn z=2 seb boot (9.2) where seb boot is the bootstrap estimate of the standard error. This interval is not accurate unless the distribution of Tn is close to Normal. Pivotal Intervals. Let = T (F ) and bn = T (Fbn ) and dene the denote bootstrap replicapivot Rn = bn . Let bn; 1 ; : : : ; bn;B tions of bn . Let H (r) denote the cdf of the pivot: H (r) = PF (Rn r): Dene Cn? = (a; b) where a = bn H 1 1 and b = bn 2 It follows that H 1 : 2 (9.4) = P(a bn bn b bn ) = P(bn b bn bn a) = P(bn b Rn bn a) = H (bn a) H (bn b) H H 1 = H H 1 1 2 2 = 1 = 1 : 2 2 Hence, Cn? is an exact 1 condence interval for . Unfortunately, a and b depend on the unknown distribution H but we P(a b) (9.3) 134 9. The Bootstrap can form a bootstrap estimate of H : Hb (r) = B 1X r) I (Rn;b B b=1 (9.5) = b where Rn;b bn . Let r denote the sample qauntile n;b ; : : : ; R ) and let denote the sample qauntile of of (Rn; 1 n;B ). Note that r = bn . It follows that an ap(n; 1 ; : : : ; n;B proximate 1 condence interval is Cn = (ba; bb) where b a = bn b b = bn b Hb 1 1 = n r1 =2 = 2bn 2 Hb 1 = bn r= 2 = 2bn 2 1 =2 : = 2 In summary: The 1 bootstrap pivotal condence interval is Cn = 2bn b 1 =2 ; 2bn b= 2 : Typically, this is a pointwise, asymptotic condence interval. Theorem 9.3 Under weak conditions on T (F ), PF (T (F ) 2 Cn) ! 1 as n ! 1, where Cn is given in (9.6). Percentile Intervals. The bootstrap percentile interval is dened by ; Cn = = 2 1 =2 : The justication for this interval is given in the appendix. Example 9.4 For estimating the skewness of the nerve data, here are the various condence intervals. (9.6) 9.3 Bootstrap Condence Intervals Method 135 95% Interval Normal (1.44, 2.09) Percentile (1.42, 2.03) Pivotal (1.48, 2.11) All these condence intervals are approximate. The probability that T (F ) is in the interval is not exactly 1 . All three intervals have the same level of accuracy. There are more accurate bootstrap condence intervals but they are more complicated and we will not discuss them here. Example 9.5 (The Plasma Cholesterol Data.) Let us return to the cholesterol data. Suppose we are interested in the dierence of the medians. Pseudo-code for the bootstrap analysis is as follows: x1 <- first sample x2 <- second sample n1 <- length(x1) n2 <- length(x2) th.hat <- median(x2) - median(x1) B <- 1000 Tboot <- vector of length B for(i in 1:B){ xx1 <- sample of size n1 with replacement from x1 xx2 <- sample of size n2 with replacement from x2 Tboot[i] <- median(xx2) - median(xx1) } se <- sqrt(variance(Tboot)) Normal <- (th.hat - 2*se, th.hat + 2*se) percentile <- (quantile(Tboot,.025), quantile(Tboot,.975)) pivotal <- ( 2*th.hat-quantile(Tboot,.975), 2*th.hat-quantile(Tboot,.025) ) The point estimate is 18.5, the bootstrap standard error is 7.42 and the resulting approximate 95 per cent condence intervals are as follows: 136 9. The Bootstrap Method 95% Interval Normal (3.7, 33.3) Percentile (5.0, 33.3) Pivotal (5.0, 34.0) Since these intervals exclude 0, it appears that the second group has higher cholesterol although there is considerable uncertainty about how much higher as reected in the width of the intervals. The next two examples are based on small sample sizes. In practice, statistical methods based on very small sample sizes might not be reliable. We include the examples for their pedagogical value but we do want to sound a note of caution about interpreting the results with some skepticism. Example 9.6 Here is an example that was one of the rst used to illustrate the bootstrap by Bradley Efron, the inventor of the bootstrap. The data are LSAT scores (for entrance to law school) and GPA. LSAT 576 651 635 605 558 653 578 575 666 545 580 572 555 594 661 GPA 3.39 3.36 3.30 3.13 2.81 3.12 3.03 2.74 3.44 2.76 3.07 2.88 3.00 3.96 3.43 Each data point is of the form Xi = (Yi; Zi ) where Yi = LSATi and Zi = GPAi . The law school is interested in the correlation RR (y Y )(z Z )dF (y; z ) = qR : R (y Y )2 dF (y ) (z Z )2 dF (z ) The plug-in estimate is the sample correlation P (Y Y )(Zi Z ) b = qP i i : P 2 2 ( Y Y ) ( Z Z ) i i i i 9.3 Bootstrap Condence Intervals 137 The estimated correlation is b = :776. The bootstrap based on B = 1000 gives seb = :137. Figure 9.1 shows the data and a histogram of the bootstrap replications b1 ; : : : ; bB . This histogram is an approximation to the sampling distribution of b. The Normalb = (:51; 1:00) based 95 per cent condence interval is :78 2se while the percentile interval is (.46,.96). In large samples, the two methods will show closer agreement. Example 9.7 This example is borrowed from An Introduction to the Bootstrap by B. Efron and R. Tibshirani. When drug companies introduce new medications, they are sometimes requires to show bioequivalence. This means that the new drug is not substantially dierent than the current treatment. Here are data on eight subjects who used medical patches to infuse a hormone into the blood. Each subject received three treatments: placebo, old-patch, new-patch. subject placebo old new old-placebo new-old ______________________________________________________________ 1 9243 17649 16449 8406 -1200 2 9671 12013 14614 2342 2601 3 11792 19979 17274 8187 -2705 4 13357 21816 23798 8459 1982 5 9055 13850 12560 4795 -1290 6 6290 9806 10157 3516 351 7 12412 17208 16570 4796 -638 8 18806 29044 26325 10238 -2719 Let Z = old placebo and Y = new old. The Food and Drug Administration (FDA) requirement for bioequivalence is that jj :20 where E (Y ) = F : E F (Z ) 9. The Bootstrap 3.4 138 • • • • 3.2 • 3.0 GPA • • • • • • 2.8 • • • • 560 580 600 620 640 660 0 50 100 150 LSAT 0.2 0.4 0.6 Bootstrap Samples FIGURE 9.1. Law school data. 0.8 1.0 9.4 Bibliographic Remarks 139 The estimate of is b = Y 452:3 = :0713: = 6342 Z The bootstrap standard error is se b = :105. To answer the bioequivalence question, we compute a condence interval. From B = 1000 bootstrap replications we get the 95 per cent interval is (-.24,.15). This is not quite contained in [-.20,.20] so at the 95 per cent level we have not demonstrated bioequivalence. Figure 9.2 shows the histogram of the bootstrap values. 9.4 Bibliographic Remarks The boostrap was invented by Efron (1979). There are several books on these topics including Efron and Tibshirani (1993), Davison and Hinkley (1997), Hall (1992), and Shao and Tu (1995). Also, see section 3.6 of van der Vaart and Wellner (1996). 9.5 Technical Appendix 9.5.1 The Jackknife There is another method for computing standard errors called the jackknife, due to Quenouille (1949). It is less computationally expensive than the bootstrap but is less general. Let Tn = T (X1 ; : : : ; Xn ) be a statistic and T( i) denote the statistic Pn th 1 with the i observation removed. Let T n = n i=1 T( i) . The jackknife estimate of var(Tn ) is n n 1X (T vjack = n i=1 ( i) T n )2 and the jackknife estimate of the standard error is se b jack = pvjack . Under suitable conditions on T , it can be shown that p vjack consistently estimates var(Tn ) in the sense that vjack =var(Tn ) ! 9. The Bootstrap 0 20 40 60 80 140 -0.3 -0.2 -0.1 0.0 Bootstrap Samples FIGURE 9.2. Patch data. 0.1 0.2 9.6 Excercises 141 1. However, unlike the bootstrap, the jackknife does not produce consistent estimates of the standard error of sample quantiles. 9.5.2 Justication For The Percentile Interval Suppose there exists a monotone transformation U = m(T ) such that U N (; c2 ) where = m(). We do not suppose we ). know the transformation, only that one exist. Let Ub = m(n;b Let u be the sample quantile of the Ub 's. Since a monotone transformation preserves quantiles, we have that u=2 = ). Also, since U N (; c2 ), the =2 quantile of U is m(= 2 z=2 c. Hence u=2 = z=2 c. Similarly, u1 =2 = + z=2 c. Therefore, P(= 2 1 =2 ) = = = m() m(1 =2 )) P(u=2 u1 =2 ) P(U cz=2 U + cz=2 ) U P( z=2 z=2 ) c ) P(m(= 2 = = 1 : An exact normalizing transformation will rarely exist but there may exist approximate normalizing transformations. 9.6 Excercises 1. Consider the data in Example 9.6. Find the plug-in estimate of the correlation coeÆcient. Estimate the standard error using the bootstrap. Find a 95 per cent condence inerval using all three methods. 2. (Computer Experiment.) Conduct a simulation to compare the four bootstrap condence interval methods. Let n = R 50 and let T (F ) = (x )3 dF (x)= 3 be the skewness. Draw Y1 ; : : : ; Xn N (0; 1) and set Xi = eYi , i = 1; : : : ; n. Construct the four types of bootstrap 95 per cent intervals for T (F ) from the data X1 ; : : : ; Xn . Repeat this whole 142 9. The Bootstrap thing many times and estimate the true coverage of the four intervals. 3. Let X1 ; : : : ; Xn t3 where n = 25. Let = T (F ) = (q:75 q:25 )=1:34 where qp denotes the pth quantile. Do a simulation to compare the coverage and length of the following condence intervals for : (i) Normal interval with standard error from the bootstrap, (ii) bootstrap percentile interval. Remark: The jackknife does not give a consistent estimator of the variance of a quantile. 4. Let X1 ; : : : ; Xn be distinct observations (no ties). Show that there are 2n 1 n distinct bootstrap samples. Hint: Imagine putting n balls into n buckets. 5. Let X1 ; : : : ; Xn be distinct observations (no ties). P Let X1 ; : : : ; Xn denote a bootstrap sample and let X n = n 1 ni=1 Xi . Find: E (X n jX1 ; : : : ; Xn), V (X n jX1 ; : : : ; Xn ), E (X n ) and V (X n ). 6. (Computer Experiment.) Let X1 ; :::; Xn Normal(; 1): Let = e and let b = eX be the mle. Create a data set (using = 5) consisting of n=100 observations. (a) Use the bootstrap to get the se and 95 percent condence interval for : (b) Plot a histogram of the bootstrap replications for the parametric and nonparametric bootstraps. These are estimates of the distribution of b. Compare this to the true sampling distribution of b. 9.6 Excercises 143 7. Let X1 ; :::; Xn Unif(0; ): The mle is b = Xmax = maxfX1 ; :::; Xng. Generate a data set of size 50 with = 1: (a) Find the distribution of b. Compare the true distribution of b to the histograms from the parametric and nonparametric bootstraps. (b) This is a case where the nonparametric bootstrap does very poorly. In fact, we can prove that this is the case. Show that, for the parametric bootstrap P (b = b) = 0 but for the nonparametric bootstrap P (b = b) :632. Hint: show that, P (b = b) = 1 (1 (1=n))n then take the limit as n gets large. 2 R 8. Let Tn =PX n , = E (X1 ), k = jx bk = n 1 ni=1 jXi X n jk . Show that jk dF (x) and 2 4X n b2 4X n b3 b4 vboot = + + 3: n n2 n 144 9. The Bootstrap 10 Parametric Inference We now turn our attention to parametric models, that is, models of the form n F = f (x; ) : 2 o (10.1) where the R k is the parameter space and = (1 ; : : : ; k ) is the parameter. The problem of inference then reduces to the problem of estimating the parameter . Students learning statistics often ask: how would we ever know that the distribution that generated the data is in some parametric model? This is an excellent question. Indeed, we would rarely hav e such knowledge which is why nonparametric methods are preferable. Still, studying methods for parametric models is useful for two reasons. First, there are some cases where background knowledge suggests that a parametric model provides a reasonable approximation. F or example, counts of traÆc accidents are known from prior experience to follow approximately a P oissonmodel. Second, the inferential concepts 146 10. Parametric Inference for parametric models provide background for understanding certain nonparametric methods. We will begin with a brief dicussion about parameters of interest and nuisance parameters in the next section, then we will discuss two methods for estimating , the method of moments and the method of maximum likelihood. 10.1 Parameter of Interest Often, we are only interested in some function T (). For example, if X N (; 2 ) then the parameter is = (; ). If our goal is to estimate then = T () is called the parameter of interest and is called a nuisance parameter. The parameter of interest can be a complicated function of as in the following example. Example 10.1 Let X1 ; : : : ; Xn Normal(; 2). The parameter is = (; ) is = f(; ) : 2 R ; > 0g. Suppose that Xi is the outcome of a blood test and suppose we are interested in , the fraction of the population whose test score is larger than 1. Let Z denote a standard Normal random variable. Then = P(X = 1 > 1) = 1 P(X P X 1 1 Z< =1 Z< : P < 1) = 1 1 < The parameter of interest is = T (; ) = 1 ((1 )= ). Example 10.2 Recall that X has a Gamma(; ) distribution if f (x; ; ) = 1 x 1 e () x= ; where ; > 0 and () = Z 1 0 y 1 e y dy x>0 10.2 The Method of Moments 147 is the Gamma function. The parameter is = (; ). The Gamma distribution is sometimes used to model lifetimes of people, animals, and electronic equipment. Suppose we want to estimate the mean lifetime. Then T (; ) = E (X1 ) = . 10.2 The Method of Moments The rst method for generating parametric estimators that we will study is called the method of moments. We will see that these estimators are not optimal but they are often easy to compute. They are are also useful as starting values for other methods that require iterative numerical routines. Suppose that the parameter = (1 ; : : : ; k ) has k components. For 1 j k, dene the j th moment j j () = E (X j ) = Z xj dF (x) (10.2) and the j th sample moment n 1X Xij : bj = n i=1 (10.3) Denition 10.3 The method of moments estimator bn is dened to be the value of such that 1 (bn ) 2 (bn ) .. . k (bn ) b1 b2 .. . = bk : = = .. . (10.4) Formula (10.4) denes a system of k equations with k unknowns. 148 10. Parametric Inference Example 10.4 Let X1 ; : : : ; Xn Bernoulli(p). Then 1 = E p (X ) = P p and b1 = n 1 n i=1 Xi . By equating these we get the estimator n 1X pbn = X: n i=1 i Example 10.5 Let X1 ; : : : ; Xn Normal(; 2 ). Then, 1 = E (X1 ) = Recall that V (X ) = and 2 = E (X12 ) = V (X1 ) + (E (X1 ))2 = 2 + 2 . We need E (X 2 ) (E (X ))2 . to solve the equations Hence, E (X 2 ) = n 1X V (X ) + (E (X ))2 . b = X n i=1 i n 1X 2 2 Xi2 : b + b = n i=1 This is a system of 2 equations with 2 unknowns. The solution is b = X n n 1X 2 (X b = n i=1 i X n )2 : Theorem 10.6 Let bn denote the method of moments estimator. Under the the conditions given in the appendix, the following statements hold: 1. The estimate bn exists with probability tending to 1. 2. The estimate is consistent: bn P! . 3. The estimate is asymptotically Normal: pn(b ) n where N (0; ) = g E (Y Y T )g T ; Y = (X; X 2 ; : : : ; X k )T , g = (g1 ; : : : ; gk ) and gj = @j 1 ()[email protected] 10.3 Maximum Likelihood 149 The last statement in the Theorem above can be used to nd standard errors and condence intervals. However, there is an easier way: the bootstrap. We defer discussion of this until the end of the chapter. 10.3 Maximum Likelihood The most common method for estimating parameters in a parametric model is the maximum likelihood method. Let X1 ; : : : ; Xn be iid with pdf f (x; ). Denition 10.7 The likelihood function is dened by Ln() = n Y i=1 f (Xi ; ): (10.5) The log-likelihood function is dened by `n () = log Ln (). The likelihood function is just the joint density of the data, except that we treat it is a function of the parameter . Thus Ln : ! [0; 1). The likelihood function is not a density function: in general, it is not true that Ln() integrates to 1. Denition 10.8 The maximum likelihood estimator , denoted by bn , is the value of that maximizes Ln(). mle The maximum of `n () occurs at the same place as the maximum of Ln (), so maximizing the log-likelihood leads to the same answer as maximizing the likelihood. Often, it is easier to work with the log-likelihood. Remark 10.9 If we multiply Ln() by any positive constant c (not depending on ) then this will not change the mle . Hence, we 10. Parametric Inference 0.4 0.6 Ln(p) 0.8 1.0 150 0.0 0.2 pbn 0.0 0.2 P p 0.4 0.6 0.8 1.0 FIGURE 10.1. Likelihood function for Bernoulli with n = 20 and n X = 12. The mle is p = 12=20 = 0:6. n i=1 i b shall often be sloppy about dropping constants in the likelihood function. Example 10.10 Suppose that X1 ; : : : ; Xn Bernoulli(p). The prob- ability function is f (x; p) = px(1 known parameter is p. Then, Ln(p) = where S = n Y f (Xi ; p) = i=1 P i Xi . n Y i=1 p)1 x pXi (1 p)1 for x = 0; 1. The unXi = pS (1 p)n S Hence, `n (p) = S log p + (n S ) log(1 p): Take the derivative of `n (p), set it equal to 0 to nd that the mle is p bn = S=n. See Figure 10.1. Example 10.11 Let X1 ; : : : ; Xn N (; 2 ). The parameter is = (; ) and the likelihood function is (ignoring some constants) Ln(; ) = Y i 1 exp 1 (X 2 2 i )2 10.3 Maximum Likelihood ( 151 ) 1 X = (Xi )2 2 2 i nS 2 n(X )2 n = exp exp 2 2 2 2 P P where X = n 1 i Xi is the sample mean and S 2 = n 1P i (Xi X )2 . The last equality above follows from the fact that Pi (Xi 2 + n(X )2 which can be veried by writing (Xi )2 = nS i P 2 2 ) = i (Xi X + X ) and then expanding the square. The log-likelihood is nS 2 n(X )2 `(; ) = n log : 2 2 2 2 Solving the equations @`(; ) @`(; ) = 0 and =0 @ @ we conclude that b = X and b = S . It can be veried that these are indeed global maxima of the likelihood. n exp Example 10.12 (A Hard Example.) Here is the example that con- fuses everyone. Let X1 ; : : : ; Xn Unif (0; ). Recall that 1 x f (x; ) = 0 0otherwise : Consider a xed value of . Suppose < Xi for some i. Then, Q f (Xi ; ) = 0 and hence Ln () = i f (Xi ; ) = 0. It follows that Ln() = 0 if any Xi > . Therefore, Ln() = 0 if < X(n) where X(n) = maxfX1 ; : : : ; Xn g. Now consider any X(n) . For every Xi we then have that f (Xi; ) = 1= so that Ln () = Q n i f (Xi ; ) = . In conclusion, Ln() = 1 n X(n) 0 < X(n) : See Figure 10.2. Now Ln() is strictly decreasing over the interval [X(n) ; 1). Hence, bn = X(n) . 0.5 0.0 0.0 0.5 1.0 x 1.5 2.0 0.0 0.5 x 1.0 =1 1.5 2.0 1.0 0.0 0.0 0.5 0.5 1.0 Ln() 1.5 1.5 = :75 f (x; ) 1.0 f (x ; ) 1.0 0.0 0.5 f (x ; ) 1.5 10. Parametric Inference 1.5 152 0.0 0.5 1.0 = 1:25 x 1.5 2.0 0.6 0.8 1.0 1.2 1.4 FIGURE 10.2. Likelihood function for Uniform (0; ). The vertical lines show the observed data. The rst three plots show f (x; ) for three dierent values of . When < X(n) = maxfX1 ; : : : ; Xn g, as in the rst plot, f (X(n) ; ) = 0 and hence Ln () = ni=1 f (Xi ; ) = 0. Otherwise f (Xi ; ) = 1= for each i and hence Ln () = ni=1 f (Xi ; ) = (1=)n . The last plot shows the likelihood function. Q Q 10.4 Properties of Maximum Likelihood Estimators. 153 10.4 Properties of Maximum Likelihood Estimators. Under certain conditions on the model, the maximum likelihood estimator bn possesses many properties that make it an appealing choice of estimator. The main properties of the mle are: (1) It is consistent: bn P! ? where ? denotes the true value of the parameter ; (2) It is equivariant: if bn is the mle of then g (bn ) is the mle of g ( ); p (3) It is asymptotically Normal: n(b ? )=seb N (0; 1) where seb can be computed analytically; (4) It is asymptotically optimal or eÆcient: roughly, this means that among all well behaved estimators, the mle has the smallest variance, at least for large samples. (5) The mle is approximately the Bayes estimator. (To be explained later.) We will spend some time explaining what these properties mean and why they are good things. In suÆciently complicated problems, these properties will no longer hold and the mle will no longer be a good estimator. For now we focus on the simpler situations where the mle works well. The properties we discuss only hold if the model satises certain regularity conditions. These are essentially smoothness conditions on f (x; ). Unless otherwise stated, we shall tacitly assume that these conditions hold. 10.5 Consistency of Maximum Likelihood Estimators. Consistency means that the mle converges in probability to the true value. To proceed, we need a denition. If f and g are 154 10. Parametric Inference 's, dene the Kullback-Leibler distance between f and g to be Z f (x) dx: (10.6) D(f; g ) = f (x) log g (x) This is not a dis- It can be shown that D(f; g ) 0 and D(f; f ) = 0. For any tance in the formal ; 2 write D(; ) to mean D(f (x; ); f (x; )). We will sense. assume that 6= implies that D(; ) > 0. Let ? denote the true value of . Maximizing `n () is equivalent to maximizing pdf Mn () = f (Xi ; ) 1X log : n i f (Xi ; ? ) By the law of large numbers, Mn () converges to E ? f (Xi ; ) log f (Xi ; ? ) = = = Z f (x; ) log f (x; ? )dx f (x; ? ) Z f (x; ? ) log f (x; ? )dx f (x; ) D(? ; ): Hence, Mn () D(? ; ) which is maximized at ? since D(? ; ? ) = 0 and D(? ; ? ) < 0 for 6= ? . Hence, we expect that the maximizer will tend to ? . To prove this formally, we need more than Mn () P! D(? ; ). We need this convergence to be uniform over . We also have to make sure that the function D(? ; ) is well behaved. Here are the formal details. Theorem 10.13 Let ? denote the true value of . Dene Mn () = f (Xi ; ) 1X log n i f (Xi ; ? ) and M () = D(? ; ). Suppose that sup jMn () M ()j P! 0 2 (10.7) 10.6 Equivariance of the mle 155 and that, for every > 0, sup M () < M (? ): :j ?j (10.8) Let bn denote the mle. Then bn P! ? . . See appendix. Proof 10.6 Equivariance of the mle Theorem 10.14 Let = g () be a one-to-one function of . Let bn be the mle of . Then bn = g (bn) is the mle of . Let h = gQ1 denote the inverse of g . Then bn = h(bn ). Q For any , L( ) = i f (xi ; h( )) = i f (xi ; ) = L() where = h( ). Hence, for ana , Ln ( ) = L() L(b) = Ln (b). Proof. Example 10.15 Let X1 ; : : : ; Xn N (; 1). The mle for is bn = X n . Let = e . Then, the mle for is b = eb = eX . 10.7 Asymptotic Normality It turns out that bn is approximately Normal and we can compute its variance analytically. To explore this, we rst need a few denitions. 156 10. Parametric Inference Denition 10.16 The score function is dened to be @ log f (X ; ) : @ The Fisher information is dened to be s(X ; ) = In () = = V n X i=1 n X i=1 (10.9) ! s(Xi ; ) V (s(Xi ; )) : (10.10) For n = 1 we will sometimes write I () instead of I1 (). It can be shown that E (s(X ; )) = 0. It then follows that V (s(X ; )) = E (s2 (X ; )). In fact a further simplication of In () is given in the next result. Theorem 10.17 In () = nI () and I () = = 2 @ log f (X ; ) E @ 2 2 Z 2 @ log f (x; ) @ 2 2 f (x; )dx: (10.11) 10.7 Asymptotic Normality Theorem 10.18 (Asymptotic Normality of the priate regularity conditions, the following hold: 1. Let se = mle .) Under appro- p 1=In (). Then, (bn se 2. Let seb = 157 q ) N (0; 1): (10.12) N (0; 1): (10.13) 1=In (bn ). Then, (bn b se ) The proof is in the appendix. The rst statementpsays that N (; se ) where the standard error of bn is se = 1=In(). The second statement says that this is still true even if we replace the standard error by its estimated standard error seb = q 1=In(bn ). Informally, the theorem says that the distribution of the mle can 2 be approximated with N (; se b ). From this fact we can construct an (asymptotic) condence interval. bn Theorem 10.19 Let Cn = bn Then, P ( 2 Cn ) ! 1 Proof. Then, P ( z=2 seb ; bn + z=2 seb : as n ! 1. Let Z denote a standard normal random variable. 2 Cn) = = P bn P bn + z=2 seb z=2 seb z=2 bn b se z=2 ! 158 10. Parametric Inference ! P( z=2 < Z < z=2 ) = 1 : For = :05, z=2 = 1:96 2, so: bn 2 seb is an approximate 95 per cent condence interval. When you read an opinion poll in the newspaper, you often see a statement like: the poll is accurate to within one point, 95 per cent of the time. They are simply giving a 95 per cent condence interval of the form bn 2seb . Example 10.20 Let X1 ; : : : ; Xn P Bernoulli(p). The mle is pbn = and f (x; p) = p)1 x , log f (x; p) = x log p + (1 x) log(1 p), s(X ; p) = (x=p) (1 x)=(1 p), and s0 (X ; p) = (x=p2 ) + (1 x)=(1 p)2 . Thus, i Xi =n px (1 p (1 p) 1 I (p) = E p ( s0 (X ; p)) = 2 + = : 2 p (1 p) p(1 p) Hence, 1 b = p se = In (pbn ) 1 pb(1 pb) 1=2 p = : n nI (pbn ) An approximate 95 per cent condence interval is pb(1 pb) 1=2 pbn 2 : n Compare this with the Hoeding interval. Example 10.21 Let X1 ; : : : ; Xn N (; 2 ) where 2 is known. The score function is s(X ; ) = (X )= 2 and s0 (X ; ) = 1= 2 so that I1 () = 1= 2 . The mle is bn = X n . According to Theorem 10.18, X n N (; 2 =n). In fact, in this case, the distribution is exact. (10.14) 10.8 Optimality 159 Poisson(). Then bn = X n and some calculations show that I1 () = 1=, so Example 10.22 Let X1 ; : : : ; Xn s 1 b = q se nI1 (bn ) Therefore,qan approximate 1 bn z=2 bn =n. = bn : n condence interval for is 10.8 Optimality Suppose that X1 ; : : : ; Xn N (; 2 ). The mle is bn = X n . Another reasonable estimator is the sample median en . The mle satises pn(b ) N (0; 2 ): n It can be proved that the median satises pn(e ) N 0; 2 : n 2 This means that the median converges to the right value but has a larger variance than the mle . More generally, consider two estimators Tn and Un and suppose that pn(T ) N (0; t2 ) n and that pn(U ) N (0; u2 ): We dene the asymptotic relative eÆciency of U to T by ARE (U; T ) = t2 =u2 . In the Normal example, ARE (en ; bn ) = 2= = :63. The interpretation is that if you use the median, you are only using about 63 per cent of the available data. Theorem 10.23 If bn is the mle and en is any other estimator then The result is acn tually more subtle than this but we needn't worry about the details. 160 10. Parametric Inference ARE (en ; bn ) 1: Thus, the mle has the smallest (asymptotic) variance and we say that mle is eÆcient or asymptotically optimal. This result is predicated upon the assumed model being correct. If the model is wrong, the mle may no longer be optimal. We will discuss optimality in more generality when we discuss decision theory. 10.9 The Delta Method. Let = g () where g is a smooth function. The maximum likelihood estimator of is b = g (b). Now we address the following question: what is the distribution of b? Theorem 10.24 (The Delta Method) If = g () where g is dier- entiable and g 0() 6= 0 then pn(b n b (b) se ) N (0; 1) (10.15) b (bn ) = jg 0 (b)j se b (bn ) se (10.16) where bn = g (bn ) and Hence, if then P ( Cn = bn z=2 seb (bn ); bn + z=2 seb (bn ) 2 Cn) ! 1 as n ! 1. Example 10.25 Let X1 ; : : : ; Xn Bernoulli(p) and let = g (p) = log(p=(1 p)). The Fisher information function is I (p) = 1=(p(1 p)) so the standard error of the mle pbn is se = fpbn (1 pbn )=ng1=2 . (10.17) 10.9 The Delta Method. 161 The mle of is b = log pb=(1 pb). Since, g 0(p) = 1=(p=(1 p)), according to the delta method 1 b ( bn ) = jg 0 (pbn )jse b (pbn ) = p : se npbn (1 pbn ) An approximate 95 per cent condence interval is 2 bn p : npbn (1 pbn ) Example 10.26 Let X1 ; : : : ; Xn N (; 2). Suppose that is known, is unknown and that we want to P estimate = log . 1 The log-likelihood is `( ) = n log 22 i (xi )2 . Dierentiate and set equal to 0 and conclude that b = P i (Xi n )2 1=2 : To get the standard error we need the Fisher information. First, log f (X ; ) = log with second derivative 1 2 and hence (X )2 2 2 3(X )2 4 1 3 2 2 + = : 2 4 2 p Hence, seb = bn = 2n: Let = g ( ) = log( ). Then, log bn . Since, g 0 = 1= , I ( ) = b ( bn ) = se bn = bn 1 b 1 p =p b 2n 2n andp an approximate 95 per cent condence interval is 2= 2n. 162 10. Parametric Inference 10.10 Multiparameter Models These ideas can directly be extended to models with several parameters. LetP = (1 ; : : : ; k ) and let b = (b1 ; : : : ; bk ) be the n mle . Let `n = i=1 log f (Xi ; ), Hjj = @ 2 `n @ 2 `n and H = : jk @j2 @j @k Dene the Fisher Information Matrix by 2 In () = E (H11 ) E (H12 ) 6 E (H ) E (H ) 22 6 21 6 . .. . 4 . . E (Hk1 ) E (Hk2 ) .. . 3 E (H1k ) E (H2k ) 7 7 7: .. 5 . E (Hkk ) (10.18) Let Jn () = In 1 () be the inverse of In . Theorem 10.27 Under appropriate regularity conditions, p n(b ) N (0; Jn ): Also, if bj is the j th component of b, then pn(b ) j j bj se N (0; 1) where seb 2j = Jn (j; j ) is the j th diagonal element of Jn . The approximate covariance of bj and bk is Cov(bj ; bk ) Jn (j; k). There is also a multiparameter delta method. Let = g (1 ; : : : ; k ) be a function and let 0 1 @g B @ C B C B .. C rg = B C . B C @ @g A @k 1 be the gradient of g . 10.11 The Parametric Bootstrap 163 Theorem 10.28 (Multiparameter delta method) Suppose that rg evaluated at b is not 0. Let b = g(b). Then pn(b ) N (0; 1) b (b) se where b (b) = se q b g )T Jbn (r b g ); (r (10.19) b g is rg evaluated at = b. Jbn = Jn (bn ) and r Example 10.29 Let X1 ; : : : ; Xn N (; 2 ). Let =. In homework question 8 you will show that n In (; ) = 2 = g (; ) = 0 0 2n2 : Hence, 2 0 2 n 0 2 1 Jn = In 1 (; ) = The gradient of g is rg = ! 2 1 : : Thus, b (b) = se q r r ( b g )T Jbn ( b g ) = 1 1 b2 1=2 pn b4 + 2b2 : 10.11 The Parametric Bootstrap For parametric models, standard errors and condence intervals may also be estimated using the bootstrap. There is only one change. In the nonparametric bootstrap, we sampled 164 10. Parametric Inference X1 ; : : : ; Xn from the empirical distribution Fbn . In the parametric bootstrap we sample instead from f (x; bn ). Here, bn could be the mle or the method of moments estimator. Example 10.30 Consider example 10.29. To get to bootstrap standardPerror, simulate X1 ; : P : : ; Xn N (b; b2 ), compute b = n 1 i Xi and b2 = n 1 i (Xi b )2 . Then compute b = g (b; b ) = b =b . Repeating this B times yields bootstrap replications b1 ; : : : ; bB and the estimated standard error is s b boot = se PB b=1 (bb B b)2 : The bootstrap is much easier than the delta method. On the other hand, the delta method has the advantage that it gives a closed form expression for the standard error. 10.12 Technical Appendix 10.12.1 Proofs Proof of Theorem 10.13. have Mn (bn ) Mn (? ). Hence, M (? ) M (bn ) = P ! Since bn maximizes Mn (), we Mn (? ) M (bn ) + M (? ) Mn (? ) Mn (b) M (bn ) + M (? ) Mn (? ) sup jMn () M ()j + M (? ) Mn (? ) 0 where the last line follows from (10.7). It follows that, for any Æ > 0, P M (bn ) < M (? ) Æ ! 0: 10.12 Technical Appendix 165 Pick any > 0. By (10.8), there exists Æ > 0 such that j ? j implies that M () < M (? ) Æ . Hence, j ? j > ) P M (bn ) < M (? ) Æ P( bn ! 0: Next we want to prove Theorem 10.18. First we need a Lemma. Lemma 10.31 The score function satises E [s(X ; )] = 0: R Note that 1 = f (x; )dx. Dierentiate both sides of this equation to conclude that Proof. @ 0 = @ = = Z Z Z Z @ f (x; )dx @ Z @f (x;) @ log f (x; ) @ f (x; )dx = f (x; )dx f ( x; ) @ f (x; )dx = s(x; )f (x; )dx = E s(X ; ): Proof of Theorem 10.18. Let `() = log L(). Then, 0 = `0 (b) `0 () + (b )`00 (): Rearrange the above equation to get b = `0 ()=`00 () or, in other words, pn(b ) = p1n `0 () 1 `00 () n TOP BOTTOM : Let Yi = @ log f (Xi ; )[email protected] Recall that E (Yi ) = 0 from the previous Lemma and also V (Yi ) = I (). Hence, X p p TOP = n 1=2 Yi = nY = n(Y i 0) W N (0; I ) 166 10. Parametric Inference @ 2 log f (Xi; )[email protected] . by the central limit theorem. Let Ai = Then E (Ai ) = I () and BOTTOM = A P! I () by the law of large numbers. Apply Theorem 6.5 part (e), to conclude that p b W d 1 n( ) = N 0; : I () I () Assuming that I () is a continuous function of , it follows that I (bn ) P! I (). Now bn = se b = pnI 1=2 (b )(b n n n ) ( pnI 1=2 ()(b )o I (bn ) n I () ) 1 =2 : The rst terms tends in distribution to N(0,1). The second term tends in probability to 1. The result follows from Theorem 6.5 part (e). Outline of Proof of Theorem 10.24. Write, b = g (b) g () + (b )g 0 () = + (b )g 0(): Thus, and hence pn(b ) pn(b )g0() p nI ()(b ) g 0 ( ) p nI ()(b ): Theorem 10.18 tells us that the right hand side tends in distribution to a N(0,1). Hence, p nI ()(b ) g 0() N (0; 1) 10.12 Technical Appendix or, in other words, where 167 b N ; se 2 (bn ) (g 0())2 : nI () The result remains true if we substitute b for by Theorem 6.5 part (e). se 2 (bn ) = 10.12.2 SuÆciency A statistic is a function T (X n) of the data. A suÆcient statistic is a statistic that contains all the information in the data. To make this more formal, we need some dentions. Denition 10.32 Write xn $ yn if f (xn; ) = c f (yn; ) for some constant c that might depend on xn and y n but not . A statistic T (xn ) is suÆcient if T (xn ) $ T (y n) implies that xn $ y n. Notice that if xn $ y n then the likelihood function based on xn has the same shape as the likelihood function based on y n. Roughly speaking, a statistic is suÆcient if we can calculate the likelihood function knowing only T (X n ). Example 10.33 Let X1 ; : : : ; Xn Example 10.34 Let X1 ; : : : ; Xn N (; ) and let T pS (1 p)n S where S = Then, f (X n; ; ) = Y i Y P Bernoulli(p). Then L(p) = i Xi so S is suÆcient. = (X; S ). f (Xi ; ; ) ( p1 exp = i 2 n 1 p = exp 2 ) 1 X (Xi )2 2 2 i nS 2 n(X )2 exp 2 2 2 2 168 10. Parametric Inference The last expression depends on the data only through T and therefore, T = (X; S ) is a suÆcient statistic. Note that U = (17 X; S ) is also a suÆcient statistic. If I tell you the value of U then you can easily gure out T and then compute the likelihood. SuÆcient statistics are far from unique. Consider the following statistics for the N (; 2 ) model: T1 (X n ) T2 (X n ) T3 (X n ) T4 (X n ) = = = = (X1 ; : : : ; Xn) (X; S ) X (X; S; X3 ): The rst statistic is just the whole data set. This is suÆcient. The second is also suÆcient as we proved above. The third is not suÆcient: you can't compute L(; ) if I only tell you X . The fourth statistic T4 is suÆcient. The statistics T1 and T4 are suÆcient but they contain redundant information. Intuitively, there is a sense in which T2 is a \more concise" suÆcient statistic than either T1 or T4 . We can express this formally by noting that T2 is a function of T1 and similarly, T2 is a function of T4 . For example, T2 = g (T4 ) where g (a1 ; a2 ; a3 ) = (a1 ; a2 ). Denition 10.35 A statistic T is minimally suÆcient if (i) it is suÆcient and (ii) it is a function of every other suÆcient statistic. Theorem 10.36 T is minimal suÆcient if T (xn ) = T (y n) if and only if xn $ y n . A statistic induces a partition on the set if outcomes. We can think of suÆciency in terms of these partitions. 10.12 Technical Appendix Example 10.37 Let X1 ; X2 ; X3 P 169 Bernoulli(). Let V = X1 , T = i Xi and U = (T; X1 ). Here is the set of outcomes and the statistics: X1 X2 V T U 0 0 0 0 (0,0) 0 1 0 1 (1,0) 1 0 1 1 (1,1) 1 1 1 2 (2,1) The partitions induced by these statistics are: V T U ! f(0; 0); (0; 1)g; f(1; 0); (1; 1)g ! f(0; 0)g; f(0; 1); (1; 0)g; f(1; 1)g ! f(0; 0)g; f(0; 1)g; f(1; 0)g; f(1; 1)g: Then V is not suÆcient but T and U are suÆcient. T is minimal suÆcient; U is not minimal since if xn = (1; 0; 1) and y n = (0; 1; 1), then xn $ y n yet U (xn ) 6= U (y n ). The statistic W = 17 T generates the same partition as T . It is also minimal suÆcient. Example 10.38 For a N (; 2 ) model, T = (X; S ) is a minimally P suÆcient statistic. For the Bernoulli model, T = i XP i is a minimally suÆcient statistic. For the Poisson model, TP= i Xi is a minimally suÆcient statistic. Check that T = ( i Xi ; X1 ) is suÆcient but not minimal suÆcient. Check that T = X1 is not suÆcient. I did not give the usual denition of suÆciency. The usual denition is this: T is suÆcient if the distribution of X n given T (X n ) = t does not depend on . Example 10.39 Two coin ips. Let X = (X1 ; X2 ) Bernoulli(p). Then T = X1 + X2 is suÆcient. To see this, we need the distribution of (X1 ; X2 ) given T = t. Since T can take 3 possible values, there are 3 conditional distributions to check. They are: 170 10. Parametric Inference (i) the distribution of (X1 ; X2 ) given T = 0: P (X1 = 0; X2 = 0jt = 0) = 1; P (X1 = 0; X2 = 1jt = 0) = 0; P (X1 = 1; X2 = 0jt = 0) = 0; P (X1 = 1; X2 = 1jt = 0) = 0 (ii) the distribution of (X1 ; X2 ) given T = 1: 1 P (X1 = 0; X2 = 0jt = 1) = 0; P (X1 = 0; X2 = 1jt = 1) = ; 2 1 P (X1 = 1; X2 = 0jt = 1) = ; P (X1 = 1; X2 = 1jt = 1) = 0 2 (iii) the distribution of (X1 ; X2 ) given T = 2: P (X1 = 0; X2 = 0jt = 2) = 0; P (X1 = 0; X2 = 1jt = 2) = 0; P (X1 = 1; X2 = 0jt = 2) = 0; P (X1 = 1; X2 = 1jt = 2) = 1: None of these depend on the parameter p. Thus, the distribution of X1 ; X2 jT does not depend on so T is suÆcient. Theorem 10.40 (Factorization Theorem) T is suÆcient if and only if there are functions g (t; ) and h(x) such that f (xn ; ) = g (t(xn ); )h(xn ). Example 10.41 Return to the two coin ips. Let t = x1 + x2 . Then f (x1 ; x2 ; ) = f (x1 ; )f (x2 ; ) = x1 (1 )1 x1 x2 (1 )1 = g (t; )h(x1 ; x2 ) where g (t; ) = t (1 )2 X1 + X2 is suÆcient. t x2 and h(x1 ; x2 ) = 1. Therefore, T = Now we discuss an implication of suÆciency in point estimation. Let b be an estimator of . The Rao-Blackwell theorem says that an estimator should only depend on the suÆcient statistic, otherwise it can be improved. Let R(; b) = E [( b)2 ] denote the mse of the estimator. 10.12 Technical Appendix 171 Theorem 10.42 (Rao-Blackwell) Let b be an estimator and let T be a suÆcient statistic. Dene a new estimator by e = E (bjT ): Then, for every , R(; e) R(; b): Example 10.43 Consider ipping a coin twice. Let b = X1 . This is a well dened (and unbiased) estimator. But it is not a function of the suÆcient statistic T = X1 + X2 . However, note that e = E (X1 jT ) = (X1 + X2 )=2. By the Rao-Blackwell Theorem, e has MSE at least as small as b = X1 . The P same applies b with n coin ips. Again dene = X1 and T = i Xi . Then P 1 e = E (X1 jT ) = n i Xi has improved mse . 10.12.3 Exponential Families Most of the parametric models we have studied so far are special cases of a general class of models called exponential families. We say that ff (x; ; 2 g is a one-parameter exponential family if there are functions (), B (), T (x) and h(x) such that f (x; ) = h(x)e()T (x) B() : It is easy to see that T (X ) is suÆcient. We call T the natural suÆcient statistic. Example 10.44 Let X Poisson(). Then x e 1 x log = e x! x! and hence, this is an exponential family with () = log , B () = , T (x) = x, h(x) = 1=x!. f (x; ) = Example 10.45 Let X Bin(n; ). Then n x f (x; ) = (1 )n x x n = exp x log + n log(1 ) : x 1 172 10. Parametric Inference In this case, () = log and 1 ; B () = n log() T (x) = x; h(x) = n : x We can rewrite an exponential family as f (x; ) = h(x)eT (x) A() where = () is called the natural parameter and A( ) = log Z h(x)eT (x) dx: For example a Poisson can be written as f (x; ) = ex e =x! where the natural parameter is = log . Let X1 ; : : : ; Xn be iid from a exponential family. Then f (xn ; ) is an exponential family: n f (xn ; ) = hn (xn )hn (xn )e()Tn (x ) Q Bn () P where hn (xn ) = i h(xi P ), Tn (xn ) = i T (xi ) and Bn () = nB (). This implies that i T (Xi ) is suÆcient. Example 10.46 Let X1 ; : : : ; Xn Uniform(0; ). Then f (xn ; ) = 1 I (x ) n (n) where I is 1 if the term inside the brackets is true and 0 otherwise, and x(n) = maxfx1 ; : : : ; xn gP . Thus T (X n ) = maxfX1 ; : : : ; Xn g is suÆcient. But since T (X n ) 6= i T (Xi ), this cannot be an exponential family. 10.12 Technical Appendix 173 Theorem 10.47 Let X have density in an exponential family. Then, E (T (X )) = A0 ( ); V (T (X )) = A00 ( ): If = (1 ; : : : ; k ) is a vector, then we say that f (x; ) has exponential family form if f (x; ) = h(x) exp ( k X j =1 ) j ()Tj (x) B () : Again, T = (T1 ; : : : ; Tk ) is suÆcient and P n iid samples also has P exponential form with suÆcient statistic ( i T1 (Xi ); : : : ; i Tk (Xi )). Example 10.48 Consider the normal family with = (; ). Now, f (x; ) = exp 2 x x2 2 2 1 2 + log(2 2 ) 2 2 : This is exponential with 1 () = 2 ; T1 (x) = x 1 ; T (x) = x2 2 () = 2 2 2 1 2 2 B () = + log(2 ) ; h(x) = 1: 2 2 P P Hence, with n iid samples, ( i Xi ; i Xi2 ) is suÆcient. As before we can write an exponential family as f (x; ) = h(x) exp T T (x) A( ) R where A( ) = h(x)eT T (x) dx. It can be shown that E (T (X )) = A_ ( ) V (T (X )) = A( ) where the rst expression is the vector of partial derivatives and the second is the matrix of second derivatives. 174 10. Parametric Inference 10.13 Exercises 1. Let X1 ; : : : ; Xn Gamma(; ). Find the method of moments estimator for and . 2. Let X1 ; : : : ; Xn Uniform(a; b) where a and b are unknown parameters and a < b. (a) Find the method of moments estimators for a and b. (b) Find the mle ba and bb. R (c) Let = xdF (x). Find the mle of . (d) Let b be the mle from (1bc).R Let e be the nonparametric plug-in estimator of = xdF (x). Suppose that a = 1, b = 3 and n = 10. Find the mse of b by simulation. Find the mse of e analytically. Compare. 3. Let X1 ; : : : ; Xn N (; 2 ). Let be the .95 percentile, i.e. P(X < ) = :95. (a) Find the mle of . (b) Find an expression for an approximate 1 condence interval for . (c) Suppose the data are: 3.23 -2.50 1.88 -0.68 4.43 1.03 -0.07 -0.01 0.76 1.76 0.33 -0.31 0.30 -0.61 1.52 1.54 2.28 0.42 2.33 -1.03 0.39 0.17 3.18 5.43 4.00 Find the mle b. Find the standard error using the delta method. Find the standard error using the parametric bootstrap. 10.13 Exercises 175 4. Let X1 ; : : : ; Xn Uniform(0; ). Show that the mle is consistent. Hint: Let Y = maxfX1 ; :::; Xn g:. For any c, P(Y < c) = P(X1 < c; X2 < c; :::; Xn < c) = P(X1 < c)P(X2 < c):::P(Xn < c). 5. Let X1 ; : : : ; Xn Poisson(). Find the method of moments estimator, the maximum likelihood estimator and the Fisher information I (). 6. Let X1 ; :::; Xn N (; 1). Dene Yi = 1 if Xi > 0 0 if Xi 0: Let = P(Y1 = 1): (a) Find the maximum likelihood estimate b of . (b) Find an approximate 95 per cent condence interval for : P (c) Dene e = (1=n) i Yi . Show that e is a consistent estimator of . (d) Compute the asymptotic relative eÆciency of e to b. Hint: Use the delta method to get the standard error of the mle . Then compute the standard error (i.e. the standard deviation) of e. (e) Suppose that the data are not really normal. Show that b is not consistent. What, if anything, does b converge to? 7. (Comparing two treatments.) n1 people are given treatment 1 and n2 people are given treatment 2. Let X1 be the number of people on treatment 1 who respond favorably to the treatment and let X2 be the number of people on treatment 2 who respond favorably. Assume that X1 Binomial(n1 ; p1 ) X2 Binomial(n2 ; p2 ). Let = p1 p2 . 176 10. Parametric Inference (a) Find the mle for . (b) Find the Fisher information matrix I (P1 ; p2 ). (c) Use the multiparameter delta method to nd the asymptotic standard error of b. (d) Suppose that n1 = n2 = 200, X1 = 160 and X2 = 148. Find b. Find an approximate 90 percent condence interval for using (i) the delta method and (ii) the parametric bootstrap. 8. Find the Fisher information matrix for Example 10.29. 9. Let X1 ; :::; Xn Normal(; 1): Let = e and let b = eX be the mle. Create a data set (using = 5) consisting of n=100 observations. (a) Use the delta method to get seb and 95 percent condence interval for . Use the parametric bootstrap to get b and 95 percent condence interval for . Use the nonse parametric bootstrap to get seb and 95 percent condence interval for : Compare your answers. (b) Plot a histogram of the bootstrap replications for the parametric and nonparametric bootstraps. These are estimates of the distribution of b. The delta method also gives b se 2 ). an approximation to this distribution namely, Normal(; Compare these to the true sampling distribution of b (whihc you can get by simulation). Which approximation, parametric bootstrap, bootstrap, or delta method is closer to the true distribution? 10. Let X1 ; :::; Xn Unif(0; ): The mle is b = X(n) = maxfX1 ; :::; Xng. Generate a data set of size 50 with = 1: (a) Find the distribution of b analytically. Compare the true distribution of b to the histograms from the parametric and nonparametric bootstraps. 10.13 Exercises 177 (b) This is a case where the nonparametric bootstrap does very poorly. Show that, for the parametric bootstrap P(b = b) = 0 but for the nonparametric bootstrap P(b = b) :632. Hint: show that, P(b = b) = 1 (1 (1=n))n then take the limit as n gets large. What is the implication of this? 178 10. Parametric Inference 11 Hypothesis Testing and p-values Suppose we want to know if a certain chemical causes cancer. We take some rats and randomly divide them into two groups. We expose one group to the chemical and then we compare the cancer rate in the two groups. Consider the following two h ypotheses: The Null Hypothesis: The cancer rate is the same in the two groups The Alternative Hypothesis: The cancer rate is not the same in the two groups. If the exposed group has a much higher rate of cancer than the unexposed group then we will reject the null hypothesis and conclude that the evidence fa vors the alternative h ypothesis; in other words we will conclude that there is evidence that the chemical causes cancer. This is an example of hypothesis testing. More formally, suppose that we partition the parameter space into two disjoint sets 0 and 1 and that we wish to test H0 : 2 0 v ersus H1 : 2 1 : (11.1) 180 11. Hypothesis Testing and p-values We call H0 the null hypothesis and H1 the alternative hypothesis. Let X be a random variable and let X be the range of X . We test a hypothesis by nding an appropriate subset of outcomes R X called the rejection region. If X 2 R we reject the null hypothesis, otherwise, we do not reject the null hypothesis: X 2 R =) reject H0 X 2= R =) retain (do not reject) H0 Usually the rejection region R is of the form n o R = x : T (x) > c where T is a test statistic and c is a critical value. The problem in hypothesis testing is to nd an appropriate test statistic T and an appropriate cuto value c. Warning! There is a tendency to use hypthesis testing methods even when they are not appropriate. Often, estimation and condence intervals are better tools. Use hypothesis testing only when you want to test a well dened hypothesis. Hypothesis testing is like a legal trial. We asume someone is innocent unless the evidence strongly suggests that he is guilty. Similarly, we retain H0 unless there is strong evidence to reject H0 . There are two types of errors when can make. Rejecting H0 when H0 is true is called a type I error. Rejecting H1 when H1 is true is called a type II error. The possible outcomes for hypothesis testing are summarized in the Table below: 11. Hypothesis Testing and p-values Retain Null p H0 true H1 true type II error 181 Reject Null type p I error Summary of outcomes of hypothesis testing. Denition 11.1 The power function of a test with rejection re- gion R is dened by () = P (X 2 R): (11.2) The size of a test is dened to be = sup (): 20 (11.3) A test is said to have level if its size is less than or equal to . A hypothesis of the form = 0 is called a simple hypothesis. A hypothesis of the form > 0 or < 0 is called a composite hypothesis. A test of the form H0 : = 0 versus H1 : 6= 0 is called a two-sided test. A test of the form H0 : 0 versus H1 : > 0 or H0 : 0 versus H1 : < 0 is called a one-sided test. The most common tests are twosided. 182 11. Hypothesis Testing and p-values N (; ) where is known. We want to test H0 : 0 versus H1 : > 0. Hence, 0 = ( 1; 0] and 1 = (0; 1). Consider the test: Example 11.2 Let X1 ; : : : ; Xn reject H0 if T > c n o where T = X . The rejection region is R = xn : T (xn ) > c . Let Z denote a standard normal random variable. The power function is () = P X > c p pn(c ) n(X ) = P > pn(c ) = P Z> p n(c ) : = 1 This function is increasing in . Hence size = sup () = (0) = 1 <0 pnc : To get a size test, set this equal to and solve for c to get c= 1 (1 ) pn : p So we reject when X > 1 (1 )= n. Equivalently, we reject when pn (X 0) > z : Finding most powerful tests is hard and, in many cases, most powerful tests don't even exist. Instead of searching for most powerful tests, we'll just consider three widely used tests: the Wald test, the 2 test and the permutation test. A fourth test, the likelihood ratio test, is dicussed in the appendix. 11.1 The Wald Test 183 11.1 The Wald Test Let be a scalar parameter, let b be an estimate of and let b be the estimated standard error of b. se The test is named after Abraham Wald (1902-1950), who was a very inuential mathematical statistician. Denition 11.3 The Wald Test Consider testing H0 : = 0 versus H1 : 6= 0 : Assume that b is asymptotically Normal: pn(b ) 0 N (0; 1): b se The size Wald test is: reject H0 when jW j > z=2 where b 0 W= : se b Theorem 11.4 Asymptotically, the Wald test has size , that is, jZ j > z=2 ! P0 as n ! 1. Under = 0 , (b 0 )=seb N (0; 1). Hence, the probability of rejecting when the null = 0 is true is Proof. P0 jW j > z=2 = P0 ! P = jb 0 j > z =2 b se jN (0; 1)j > z=2 : ! Remark 11.5 Most texts dene the Wald test slightly dierently. They use the standard error computed at = 0 rather then at the estimated value b. Both versions are valid. 184 11. Hypothesis Testing and p-values Let us consider the power of the Wald test when the null hypothesis is false. Theorem 11.6 Suppose the true value of is ? = 6 0 . The power (? ) { the probability of correctly rejecting the null hypothesis { is given (approximately) by 1 0 b se ? + z=2 + 0 b se ? z=2 : (11.4) Recall that seb tends to 0 as the sample size increases. Inspecting (11.4) closely we note that: (i) the power is large if ? is far from 0 and (ii) the power is large if the sample size is large. Example 11.7 (Comparing Two Prediction Algorithms) We test a prediction algorithm on a test set of size m and we test a second prediction algorithm on a second test set of size n. Let X be the number of incorrect predictions for algorithm 1 and let Y be the number of incorrect predictions for algorithm 2. Then X Binomial(m; p1 ) and Y Binomial(n; p2 ). To test the null hypothesis that p1 = p2 write H0 : Æ = 0 versus H1 : Æ 6= 0 where Æ = p1 standard error p2 . The r b = se mle is Æb = pb1 pb2 with estimated pb1 (1 pb1 ) pb2 (1 pb2 ) + : m n The size Wald test is to reject H0 when jW j > z=2 where W= Æb 0 b se = q pb1 pb1 (1 pb1 ) m pb2 + pb2 (1 pb2 ) n : The power of this test will be largest when p1 is far from p2 and when the sample sizes are large. 11.1 The Wald Test 185 What if we used the same test set to test both algorithms? The two samples are no longer independent. Instead we use the following strategy. Let Xi = 1 if algorithm 1 is correct on test case i and Xi = 0 otherwise. Let Yi = 1 if algorithm 2 is correct on test case i Yi = 0 otherwise. A typical data set will look something like this: Test Case Xi Yi Di = Xi 1 2 3 4 5 .. . n 1 1 1 0 0 .. . 0 0 1 1 1 0 .. . 1 1 0 0 -1 0 .. . -1 Yi Let Æ = E (Di ) = E (Xi ) E (Yi ) = P(Xi = 1) P(Yi = 1): p P ThenPÆb = D = n 1 ni=1 Di and seb (Æb) = S= n where S 2 = n 1 ni=1 (Di D)2 . To test H0 : Æ = 0 versus H1 : Æ 6= 0 we use b se b and reject H0 if jW j > z=2 . This is called a paired W = Æ= comparison. Example 11.8 (Comparing Two Means.) Let X1 ; : : : ; Xm and Y1 ; : : : ; Yn be two independent samples from populations with means 1 and 2 , respectively. Let's test the null hypothesis that 1 = 2 . Write this as H0 : Æ = 0 versus H1 : Æ 6= 0 where Æ = 1 2 . Recall that the nonparametric plug-in estimate of Æ is Æb = X Y with estimated standard error r b = se s21 s22 + m n 186 11. Hypothesis Testing and p-values where s21 and s22 are the sample variances. The size Wald test rejects H0 when jW j > z=2 where W= Æb 0 b se = X q Y s2 m+ n s21 2 : Example 11.9 (Comparing Two Medians.) Consider the previous example again but let us test whether the medians of the two distributions are the same. Thus, H0 : Æ = 0 versus H1 : Æ 6= 0 where Æ = 1 2 where 1 and 2 are the medians. The nonparametric plug-in estimate of Æ is Æb = b1 b2 where b1 and b2 are the sample medians. The estimated standard error seb of Æb can be obtained from the bootstrap. The Wald test statistic is b se b. W = Æ= There is a relationship between the Wald test and the 1 asymptotic condence interval b seb z=2 . Theorem 11.10 The size Wald test rejects H0 : = 0 versus H1 : 6= 0 if and only if 0 2= C where C = (b seb z=2 ; b + seb z=2 ): Thus, testing the hypothesis is equivalent to checking whether the null value is in the condence interval. 11.2 p-values Reporting \reject H0 " or \retain H0 " is not very informative. Instead, we could ask, for every , whether the test rejects at that level. Generally, if the test rejects at level it will also reject at level 0 > . Hence, there is a smallest at which the test rejects and we call this number the p-value. 11.2 p-values 187 Denition 11.11 Suppose that for every 2 (0; 1) we have a size test with rejection region R . Then, n p-value = inf : T (X n) o 2 R : That is, the p-value is the smallest level at which we can reject H0 . Informally, the p-value is a measure of the evidence against H0 : the smaller the p-value, the stronger the evidence against H0 . Typically, researchers use the following evidence scale: p-value < :01 .01 - .05 .05 - .10 > :1 evidence very strong evidence againt H0 strong evidence againt H0 weak evidence againt H0 little or no evidence againt H0 Warning! A large p-value is not strong evidence in favor of H0 . A large p-value can occur for two reasons: (i) H0 is true or (ii) H0 is false but the test has low power. But do not confuse the p-value with P(H0 jData). The pvalue is not the probability that the null hypothesis is true. The following result explains how to compute the p-value. Theorem 11.12 Suppose that the size test is of the form reject H0 if and only if T (X n) c : Then, p-value = sup P (T (X n ) T (xn )): 20 In words, the p-value is the probability (under H0 ) of observing a value of the test statistic as or more extreme than what was actually observed. We discuss quantities like P(H0 jData) in the chapter on Bayesian inference. 188 11. Hypothesis Testing and p-values For a Wald test, W has an aproximate N (0; 1) distribution under H0 . Hence, the p-value is p-value P jZ j > jwj = 2P Z < jw j = 2 Z < jw j (11.5) 0 )=seb is the observed value of where Z N (0; 1) and w the test statistic. Here is an important property of p-values. = (b Theorem 11.13 If the test statistic has a continuous distribu- tion, then under H0 : = 0 , the p-value has a Uniform (0,1) distribution. If the p-value is less than .05 then people often report that \the result is statistically signicant at the 5 per cent level." This just means that the null would be rejected if we used a size = 0:05 test. In general, the size test rejects if and only if p-value < . Example 11.14 Recall the cholesterol data from Example 8.11. To test of the means ard dierent we compute W= Æb 0 b se = X q Y s21 m + sn2 2 = 216:2 195:3 = 3:78: 52 + 2:42 To compute the p-value, let Z N (0; 1) denote a standard Normal random variable. Then, p-value = P(jZ j > 3:78) = 2P(Z < 3:78) = :0002 which is very strong evidence against the null hypothesis. To test if the medians are dierent, let b1 and b2 denote the sample medians. Then, b W= 1 b se b2 = 212:5 194 = 2:4 7:7 11.3 The 2 distribution 189 where the standard error 7.7 was found using the boostrap. The p-value is p-value = P(jZ j > 2:4) = 2P(Z < 2:4) = :02 which is strong evidence against the null hypothesis. Warning! A result might be statistically signicant and yet the size of the eect might be small. In such a case we have a result that is statistically signicant but not practically signicant. It is wise to report a condence interval as well. 11.3 The distribution 2 Let Z ; : : : ; Zk be independent, standard normals. Let V = Then we say that V has a 2 distribution with k degrees of freedom, written V 2k . The probability density of V is v (k=2) 1 e v=2 f (v ) = k=2 2 (k=2) for v > 0. It can be shown that E (V ) = k and V (k) = 2k. We dene the upper quantile 2k; = F 1 (1 ) where F is the 2 2 cdf . That is, P (k > k; ) = . 1 Pk 2. Z i=1 i 11.4 Pearson's Test For Multinomial Data 2 Pearson's 2 test is used for multinomial data. Recall that X = (X1 ; : : : ; Xk ) has a multinomial distribution if n f (x1 ; : : : ; xk ; p) = px1 pxkk x1 : : : xk 1 where n n! = : x1 : : : xk x1 ! xk ! The mle of p is pb = (pb1 ; : : : ; pbk ) = (X1 =n; : : : ; Xk =n). 190 11. Hypothesis Testing and p-values Let (p01 ; : : : ; p0k ) be some xed set of probabilities and suppose we want to test H0 : (p1 ; : : : ; pk ) = (p01 ; : : : ; p0k ) versus H1 : (p1 ; : : : ; pk ) 6= (p01 ; : : : ; p0k ): Pearson's 2 statistic is k X (Xj k np0j )2 X (Oj Ej )2 T= = np0j Ej j =1 j =1 where Oj = Xj is the observed data and Ej = E (Xj ) = np0j is the expected value of Xj under H0 . 2k 1 . Hence the test: reject H0 if T > 2k 1; has asymptotic level . The p-value is P(2k > t) where t is the observed value of the test statistic. Theorem 11.15 Under H0 , T Example 11.16 (Mendel's peas.) Mendel bred peas with round yel- low seeds and wrinkled green seeds. There are four types of progeny: round yellow, wrinkled yellow, round green and (4) wrinkled green. The number of each type is multinomial with probability (p1 ; p2 ; p3 ; p4 ). His theory of inheritance predicts that 9 3 3 1 p= ; ; ; p0 : 16 16 16 16 In n = 556 trials he observed X = (315; 101; 108; 32). Since, np10 = 312:75; np20 = np30 = 104:25 and np40 = 34:75, the test statistic is (315 312:75)2 (101 104:25)2 (108 104:25)2 (32 34:75)2 + + + = 0:47: 2 = 312:75 104:25 104:25 34:75 The = :05 value for a 23 is 7.815. Since 0:47 is not larger than 7.815 we do not reject the null. The p-value is p-value = P(23 > :47) = :07 which is only moderate evidence againt H0 . Hence, the data do not contradict Mendel's theory. Interestingly, there is some controversey about whether Mendel's results are \too good." 11.5 The Permutation Test 191 11.5 The Permutation Test The permutation test is a nonparametric method for testing whether two distribution are the same. This test is \exact" meaning that it is not based on large sample theory approximations. Suppose that X1 ; : : : ; Xm FX and Y1 ; : : : ; Yn FY are two independent samples and H0 is the hypothesis that the two samples are identically distributed. This is the type of hypothesis we would consider when testing whether a treatment diers from a placebo. More precisely we are testing H0 : FX = FY versus H1 : FX = 6 FY : Let T (x1 ; : : : ; xm ; y1 ; : : : ; yn) be some test statistic, for example, T (X1 ; : : : ; Xm ; Y1; : : : ; Yn ) = jX m Y n j: Let N = m + n and consider forming all N ! permutations of the data X1 ; : : : ; Xm ; Y1 ; : : : ; Yn . For each permutation, compute the test statistic T . Denote these values by T1 ; : : : ; TN ! . Under the null hypothesis, each of these values is equally likely. The distribution P0 that puts mass 1=N ! on each Tj is called the permutation distribution of T . Let tobs be the observed value of the test statistic. Assuming we reject when T is large, the pvalue is N! 1 X p-value = P0 (T > tobs ) = I (T > t ): N ! j =1 j obs Example 11.17 Here is a toy example to make the idea clear. Suppose the data are: (X1 ; X2 ; Y1 ) = (1; 9; 3). Let T (X1 ; X2 ; Y1) = jX Y j = 2. The permutations are: Under the null hypothesis, given the ordered data values, X1 ; : : : ; Xm ; Y1; : : : ; Yn is uniformly distributed over the N ! permutations of the data. 192 11. Hypothesis Testing and p-values permutation value of T probability (1,9,3) 2 1/6 (9,1,3) 2 1/6 (1,3,9) 7 1/6 (3,1,9) 7 1/6 (3,9,1) 5 1/6 (9,3,1) 5 1/6 The p-value is P(T > 2) = 4=6. Usually, it is not practical to evaluate all N ! permutations. We can approximate the p-value by sampling randomly from the set of permutations. The fraction of times Tj > tobs among these samples approximates the p-value. Algorithm for Permutation Test 1. Compute the observed value of the test statistic tobs = T (X1 ; : : : ; Xm ; Y1 ; : : : ; Yn). 2. Randomly permute the data. Compute the statistic again using the permuted data. 3. Repeat the previous step B times and let T1 ; : : : ; TB denote the resulting values. 4. The approximate p-value is B 1X I (T > t ): B j =1 j obs Example 11.18 DNA microarrays allow researchers to measure the expression levels of thousands of genes. The data are the levels of messenger RNA (mRNA) of each gene, which is thought 11.6 Multiple Testing 193 to provide a measure how much protein that gene produces. Roughly, the larger the number, the more active the gene. The table below, reproduced from Efron, et. al. (JASA, 2001, p. 1160) shows the expression levels for genes from two types of liver cancer cells. There are 2,638 genes in this experiment but here we show just the rst two. The data are log-ratios of the intensity levels of two dierent color dyes used on the arrays. Type I Type II Patient 1 2 3 4 5 6 7 8 9 10 Gene 1 230.0 -1,350 -1,580.0 -400 -760 970 110 -50 -190 -200 Gene 2 470.0 -850 -.8 -280 120 390 -1730 -1360 -.8 -330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Let's test whether the median level of gene 1 is dierent between the two groups. Let 1 denote the median level of gene 1 of Type I and let 2 denote the median level of gene 1 of Type II. The absolute dierence of sample medians is T = jb1 b2 j = 710. Now we estimate the permutation distribution by simulation and we nd that the estimated p-value is .045. Thus, if we use a = :05 level of signicance, we would say that there is evidence to reject the null hypothesis of no dierence. In large samples, the permutation test usually gives similar results to a test that is based on large sample theory. The permutation test is thus most useful for small samples. 11.6 Multiple Testing In some situations we may conduct many hypothesis tests. In example 11.18, there were actually 2,638 genes. If we tested for a dierence for each gene, we would be conducting 2,638 separate hypothesis tests. Suppose each test is conducted at level . For any one test, the chance of a false rejection of the null is . But the chance of at least one false rejection is much higher. 194 11. Hypothesis Testing and p-values This is the multiple testing problem. The problem comes up in many data mining situations where one may end up testing thousands or even millions of hypotheses. There are many ways to deal with this problem. Here we discuss two methods. Consider m hypothesis tests: H0i versus H1i ; i = 1; : : : ; m and let P1 ; : : : ; Pm denote m p-values for these tests. The Bonferroni Method Given p-values P1 ; : : : ; Pm , reject null hypothesis H0i if Pi < =m. Theorem 11.19 Using the Bonferroni method, the probability of falsely rejecting any null hypotheses is less than or equal to . Let R be the event that at least one null hypotheses is falsely rejected. Let Ri be the event that the ith null hypothesisSis falsely rejected. Recall that if A1 ; : : : ; Ak are events then Pk k P( i=1 Ai ) i=1 P(Ai ). Hence, Proof. P(R) = P m [ i=1 ! Ri m X i=1 P (Ri ) = m X = m i=1 from Theorem 11.13. Example 11.20 In the gene example, using = :05, we have that :05=2638 = :00001895375. Hence, for any gene with p-value less than .00001895375, we declare that there is a signicant dierence. The Bonferroni method is very conservative because it is trying to make it unlikely that you would make even one false rejection. Sometimes, a more reasonable idea is to control the 11.6 Multiple Testing 195 false discovery rate (FDR) which is dened as the mean of the number of false rejections divided by the number of rejections. Suppose we reject all null hypotheses whose p-values fall below some threshold. Let m0 be the number of null hypotheses are true and m1 = m m0 null hypotheses are false. The tests can be categorize in a 2 2 as in the following table. H0 True H0 False Total H0 Not Rejected H0 Rejected U V T S m R R Total m0 m1 m Dene the False Discovery Proportion (FDP) FDP = V=R if R > 0 0 if R = 0: The FDP is the proportion of rejections that are incorrect. Next dene FDR = E (FDP). The Benjamini-Hochberg (BH) Method 1. Let P(1) < < P(m) denote the ordered p-values. 2. Dene `i = i ; Cm m and n R = max i : P(i) < `i o (11.6) where Cm isPdened to be 1 if the p-values are independent and Cm = m i=1 (1=i) otherwise. 3. Let t = P(R) ; we call t the BH rejection threshold. 4. Rejects all null hypotheses H0i for which Pi t: 196 11. Hypothesis Testing and p-values Theorem 11.21 (Benjamini and Hochberg) If the procedure above is applied, then regardless of how many nulls are true and regardless of the distribution of the p-values when the null hypthesis is false, FDR = E (FDP) m0 : m Example 11.22 Figure 11.1 shows 7 ordered p-values plotted as vertical lines. If we tested at level without doing any corection for mutiple testing, we would reject all hypotheses whose p-values are less than . In this case, the 5 hypotheses corresponding to the 5 smallest p-values are rejected. The Bonferroni method rejects all hypotheses whose p-values are less than =m. In this example, this leads to no rejections. The BH threshold corresponds to the last p-value that falls under the line with slope . This leads to three hypotheses being rejected in this case. The Bonferonni method controls the probability of a single false rejection. This is very strict and leads to low power when there are many tests. The FDR method controls the fraction of false discoveries which is a more reasonable criterion when there are many tests. Summary. 11.7 Technical Appendix 11.7.1 The Neyman-Pearson Lemma In the special case of a simple null H0 : = 0 and a simple alternative H1 : = 1 we can say precisely what the most powerful test is. Theorem 11.23 (Neyman-Pearson.) Suppose we test H0 : = 0 versus H1 : = 1 . Let Qn f (x ; ) L (1 ) T= = Qni=1 i 1 : L(0) i=1 f (xi ; 0 ) 11.7 Technical Appendix 197 p-values t =m 0 reject don't reject threshold FIGURE 11.1. Schematic illustration of Benjamini-Hochberg procedure. All hypotheses corresponding to the last undercrossing are rejected. 1 198 11. Hypothesis Testing and p-values Suppose we reject H0 when T > k. If we choose k so that P0 (T > k ) = then this test is the most powerful, size test. That is among all tests with size , this test maximizes the power (1 ). 11.7.2 Power of the Wald Test 11.6. Let Z N (0; 1). Then, Proof of Theorem Power = (? ) = P? (Reject H0 ) = P? (jW j > z=2 ) ! j b 0 j = P ? > z=2 b se = = = P ? b 0 b se ! > z=2 + P? b 0 b se ! < z=2 b z=2 + P? b < 0 se b z=2 P? b > 0 + se ! b ? 0 ? b ? 0 ? > + z=2 + P? < P ? b b b b se se se se 0 ? + z=2 + P Z < 0 ? z=2 P Z> b b se se = 1 0 b se ? + z=2 + 0 b se ? z=2 : 11.7.3 The t-test To test H0 : = 0 where is the mean, we can use the Wald test. When the data are assumed to be Normal and the sample size is small, it is common instead to use the t-test. A random variable T has a t-distribution with k degrees of freedom if it has ! z=2 11.7 Technical Appendix density f (t) = p k k+1 2 k t2 2 1+ k 199 (k+1)=2 : When the degrees of freedom k ! 1, this tends to a Normal distribution. When k = 1 it reduces to a Cauchy. Let X1 ; : : : ; Xn N (; 2 ) where = (; 2 ) are both unknown. Suppose we want to test = 0 versus 6= 0 . Let T= pn(X n Sn 0 ) where Sn2 is the sample variance. For large samples T N (0; 1) under H0 . The exact distribution of T under H0 is tn 1 . Hence if we reject when jT j > tn 1;=2 then we get a size test. 11.7.4 The Likelihood Ratio Test Let = (1 ; : : : ; q ; q+1 ; : : : ; r ) and suppose that 0 consists of all parameter values such that (q+1 ; : : : ; r ) = (0;q+1 ; : : : ; 0;r ). Denition 11.24 Dene the likelihood ratio statistic by sup2 L() = 2 log = 2 log sup20 L() L(b) L(b0) ! where b is the mle and b0 is the mle when is restricted to lie in 0 . The likelihood ratio test is: reject H0 when (xn ) > 2r q; . For example, if = (1 ; 2 ; 3 ; 4 ) and we want to test the null hypothesis that 3 = 4 = 0 then the limiting distribution has 4 2 = 2 degrees of freedom. 200 11. Hypothesis Testing and p-values Theorem 11.25 Under H0 , d 2 2 log (xn ) ! r q : Hence, asymptotically, the LR test is level . 11.8 Bibliographic Remarks The most complete book on testing is (1986). See also Casella and Berger (1990, Chapter 8). The FDR method is due to Benjamini and Hochberg (1995). 11.9 Exercises 1. Prove Theorem 11.13. 2. Prove Theorem 11.10. 3. Let X1 ; :::; Xn Uniform(0; ) and let Y = maxfX1 ; :::; Xn g. We want to test H0 : = 1=2 versus H1 : > 1=2. The Wald test is not appropriate since Y does not converge to a Normal. Suppose we decide to test this hypothesis by rejecting H0 when Y > c. (a) Find the power function. (b) What choice of c will make the size of the test .05? (c) In a sample of size n = 20 with Y=0.48 what is the p-value? What conclusion about H0 would you make? (d) In a sample of size n = 20 with Y=0.52 what is the p-value? What conclusion about H0 would you make? 4. There is a theory that people can postpone their death until after an important event. To test the theory, Phillips 11.9 Exercises 201 and King (1988) collected data on deaths around the Jewish holiday Passover. Of 1919 deaths, 922 died the week before the holiday and 997 died the week after. Think of this as a binomial and test the null hypothesis that = 1=2: Report and interpret the p-value. Also construct a condence interval for : Reference: Phillips, D.P. and King, E.W. (1988). Death takes a holiday: Mortality surrounding major social occasions. The Lancet, 2, 728-732. 5. In 1861, 10 essays appeared in the New Orleans Daily Crescent. They were signed \Quintus Curtuis Snodgrass" and some people suspected they were actually written by Mark Twain. To investigate this, we will consider the proportion of three letter words found in an author's work. From eight Twain essays we have: .225 .262 .217 .240 .230 .229 .235 .217 From 10 Snodgrass essays we have: .209 .205 .196 .210 .202 .207 .224 .223 .220 .201 (source: Rice xxxx) (a) Perform a Wald test for equality of the means. Use the nonparametric plug-in estimator. Report the p-value and a 95 per cent condence interval for the dierence of means. What do you conclude? (b) Now use a permutation test to avoid the use of large sample methods. What is your conclusion? 6. Let X1 ; : : : ; Xn N (; 1). Consider testing H0 : = 0 versus = 1: 202 11. Hypothesis Testing and p-values Let the rejection region be R = fxn : T (xn ) > cg where P T (xn ) = n 1 ni=1 Xi . (a) Find c so that the test has size . (b) Find the power under H1 , i.e. nd (1). (c) Show that (1) ! 1 as n ! 1. 7. Let b be the mle of a parameter and let seb = fnI (b)g 1=2 where I () is the Fisher information. Consider testing H0 : = 0 versus 6= 0 : Consider the Wald test with rejection region R = fxn : jZ j > z=2 g where Z = (b 0 )=seb . Let 1 > 0 be some alternative. Show that (1 ) ! 1. 8. Here are the number of elderly Jewish and Chinese women who died just before and after the Chinese Harvest Moon Festival. Week Chinese Jewish -2 55 141 33 145 -1 1 70 139 2 49 161 Compare the two mortality patterns. 9. A randomized, double-blind experiment was conducted to assess the eectiveness of several drugs for reducing postoperative nausea. The data are as follows. Number of Patients Incidence of Nausea Placebo 80 45 Chlorpromazine 75 26 Dimenhydrinate 85 52 Pentobarbital (100 mg) 67 35 Pentobarbital (150 mg) 85 37 11.9 Exercises 203 (source: ) (a) Test each drug versus the placebo at the 5 per cent level. Also, report the estimated odds-ratios. Summarize your ndings. (b) Use the Bonferoni and the FDR method to adjust for multiple testing. 10. Let X1 ; :::; Xn Poisson(): (a) Let 0 > 0. Find the size Wald test for H0 : = 0 versus H1 : 6= 0 : (b) (Computer Experiment.) Let 0 = 1, n = 20 and = :05. Simulate X1 ; : : : ; Xn Poisson(0 ) and perform the Wald test. Repeat many times and count how often you reject the null. How close is the type I error rate to .05? 204 11. Hypothesis Testing and p-values 12 Bayesian Inference 12.1 The Bay esianPhilosophy The statistical theory and methods that we hav e discussed so far are known as frequentist (or classical) inference. The frequentist point of view is based on the following postulates: (F1) Probability refers to limiting relative frequencies. Probabilities are objective properties of the real world. (F2) P arameters are xed, (usually unknown) constants. Because they are not uctuating, no probability statements can be made about parameters. (F3) Statistical procedures should be designed to hav e well dened long run frequency properties. F or example, a 95 per cent condence in terval should trap the true v alue of the parameter with limiting frequency at least 95per cent. There is another approach to inference called Bay esian inference. The Bay esian approach is based on the following postulates: 206 12. Bayesian Inference (B1) Probability describes degree of belief, not limiting frequency. As such, we can make probability statements about lots of things, not just data which are subject to random variation. For example, I might say that `the probability that Albert Einstein drank a cup of tea on August 1 1948" is .35. This does not refer to any limiting frequency. It reects my strength of belief that the proposition is true. (B2) We can make probability statements about parameters, even though they are xed constants. (B3) We make inferences about a parameter , by producing a probability distribution for . Inferences, such as point estimates and interval estimates may then be extracted from this distribution. Bayesian inference is a controversial approach because it inherently embraces a subjective notion of probability. In general, Bayesian methods provide no guarantees on long run performance. The eld of Statistics puts more emphasis on frequentist methods although Bayesian methods certainly have a presence. Certain data mining and machine learning communuties seem to embrace Bayesian methods very strongly. Let's put aside philosophical arguments for now and see how Bayesian inference is done. We'll conclude this chapter with some discussion on the strengths and weaknesses of each approach. 12.2 The Bayesian Method Bayesian inference is usually carried out in the following way. 1. We choose a probability density f () { called the prior distribution { that expresses our degrees of beliefs about a parameter before we see any data. 12.2 The Bayesian Method 207 2. We choose a statistical model f (xj) that reects our beliefs about x given . Notice that we now write this as f (xj) instead of f (x; ). 3. After observing data X1 ; : : : ; Xn, we update our beliefs and form the posterior distribution f (jX1 ; : : : ; Xn ). To see how the third step is carried out, rst, suppose that is discrete and that there is a single, discrete observation X . We should use a capital letter now to denote the parameter since we are treating it like a random variable so let denote the parameter. Now, in this discrete setting, P (X = x; = ) PP (X = xj = )P ( = ) = P ( = jX = x) = P (X = x) P (X = xj = )P ( = ) which you may recognize from earlier in the course as Bayes' theorem. The version for continuous variables is obtained by using density functions: f (xj)f () f (jx) = R : (12.1) f (xj)f ()d If we have n iid observations X1 ; : : : ; Xn, we replace f (xj) Qn with f (x1 ; : : : ; xn j) = i=1 f (xi j). Let us write X n to mean (X1 ; : : : ; Xn ) and xn to mean (x1 ; : : : ; xn ). Then L ()f () / L ()f (): f (xn j)f () =R n f (jxn ) = R n f (x j)f ()d Ln()f ()d n (12.2) In the right hand side of the last equation, we threw away the R denominator Ln()f ()d which is a constant that does not depend on ; we call this quantity the normalizing constant. We can summarize all this by writing: \posterior is proportional to likelihood times prior:00 (12.3) You might wonder, doesn't it cause a problem to throw away R the constant Ln ()f ()d? The answer is that we can always 208 12. Bayesian Inference R recover the constant is since we know that f (jxn )d = 1. Hence, we often omit the constant until we really need it. What do we do with the posterior? First, we can get a point estimate by summarizing the center of the posterior. Typically, we use the mean or mode of the posterior. The posterior mean is R Z L ()f () n n = f (jx )d = R n (12.4) Ln()f ()d : WeRcan also obtain aRBayesian interval estimate. Dene a and b by a1 f (jxn )d = b1 f (jxn )d = =2. Let C = (a; b). Then P( so C is a 1 2 Cj xn ) = Z b a f (jxn ) d = 1 posterior interval. Example 12.1 Let X1 ; : : : ; Xn Bernoulli(p). Suppose we take the uniform distribution f (p) = 1 as a prior. By Bayes' theorem the posterior has the form f (pjxn ) / f (p)Ln(p) = ps (1 p)n s = ps+1 1 (1 p)n s+1 1 P where s = i xi is the number of heads. Recall that random variable has a Beta distribution with parameters and if its density is f (p; ; ) = ( + ) 1 p (1 p) 1 : () ( ) We see that the posterior for p is a Beta distribution with parameters s + 1 and n s + 1. That is, f (pjxn ) = (n + 2) p(s+1) 1 (1 p)(n s+1) 1 : (s + 1) (n s + 1) We write this as pjxn Beta(s + 1; n s + 1): 12.2 The Bayesian Method 209 Notice that we have gured Rout the normalizing constant without actually doing the integral Ln (p)f (p)dp. The mean of a Beta (; ) is =( + ) so the Bayes estimator is p= s+1 : n+2 It is instructive to rewrite the estimator as p = n pb + (1 n)pe where pb = s=n is the mle, pe = 1=2 is the prior mean and n = n=(n + 2) 1. A 95 per cent posterior interval can be obtained Rb by numerically nding a and b such that a f (pjxn ) dp = :95. Suppose that instead of a uniform prior, we use the prior p Beta(; ). If you repeat the calculations above, you will see that pjxn Beta( + s; + n s). The at prior is just the special case with = = 1. The posterior mean is +s n + p= = pb + p ++n ++n ++n 0 where p0 = =( + ) is the prior mean. In the previous example, the prior was a Beta distribution and the posterior was a Beta distribution. When the prior and the posterior are in the same family, we say that the prior is conjugate. N (; 2 ). For simplicity, let us assume that is known. Suppose we take as a prior N (a; b2 ). Example 12.2 Let X1 ; : : : ; Xn In problem 1 in the homework, it is shown that the posterior for is jX n N (a; b2 ) (12.5) where = wX + (1 w)a 210 12. Bayesian Inference where 1 1 1 1 w = 1 se2 1 and 2 = 2 + 2 se b se2 + b2 p and se = = n is the standard error of the mle X . This is another example of a conjugate prior. Note that w ! 1 and =se ! 1 as n ! 1. So, for large n, the posterior is approxb se2 ). The same is true if n is xed but b ! 1, imately N (; which corresponds to letting the prior become very at. Continuing with this example, let is nd C = (c; d) such that P r( 2 C jX n) = :95. We can do this by choosing c such that P r( < cjX n ) = :025 and P r( > djX n) = :025. So, we want to nd c such that c n n P ( < cjX ) = P < X c = P Z< = :025: Now, we know that P (Z < 1:96) = :025. So c = 1:96 implying that c = 1:96: By similar arguments, d = + 1:96: So a 95 per cent Bayesian interval is 1:96 . Since b and se, the 95 per cent Bayesian interval is approximated by b 1:96 se which is the frequentist condence interval. 12.3 Functions of Parameters How do we make inferences about a function = g ()? Remember in Chapter 3 we solved the following problem: given the density fX for X , nd the density for Y = g (X ). We now simply apply the same reasoning. The posterior cdf for is H ( j xn ) = P(g () ) = Z A f (jxn )d 12.4 Simulation 211 where A = f : g () g. The posterior density is h( jxn ) = H 0 ( jxn ). Example 12.3 Let X1 ; : : : ; Xn that pj log(p=(1 Xn H( j xn) Bernoulli(p) P and f (p) = 1 so Beta(s + 1; n s + 1) with s = n i=1 xi . p)). Then = = = = n x ) = P log 1 e n x 1+e j P( P P Z e =(1+e ) 0 P P Let = n x f (pjxn) dp Z e =(1+e ) (n + 2) ps(1 (s + 1) (n s + 1) 0 p)n s dp and h( jxn ) = H 0 ( jxn ) = = = for s s (n + 2) e (s + 1) (n s + 1) 1 + e e (n + 2) (s + 1) (n s + 1) 1 + e (n + 2) e (s + 1) (n s + 1) 1 + e 1 1+e 1 1+e s 1 1+e n s 0 @ 1+e e @ @ n s n s+2 1 1+e 2 R. 12.4 Simulation The posterior can often be approximated by simulation. Suppose we draw 1 ; : : : ; B p(jxn ). Then a histogram of 1 ; : : : ; B approximates the posterior density p(jxn ). An approximation 1 2 A 212 12. Bayesian Inference P to the posterior mean n = E (jxn ) is B 1 Bj=1 j : The posterior 1 interval can be approximated by (=2 ; 1 =2 ) where =2 is the =2 sample quantile of 1 ; : : : ; B . Once we have a sample 1 ; : : : ; B from f (jxn ), let i = g (i ). Then 1 ; : : : ; B is a sample from f ( jxn ). This avoids the need to do any analytical calculations. Simulation is discussed in more detail later in the book. Example 12.4 Consider again Example 12.3. We can approximate the posterior for without doing any calculus. Here are the steps: 1. Draw P1 ; : : : ; PB Beta(s + 1; n s + 1). 2. Let i = log(Pi =(1 Pi )) for i = 1; : : : ; B . Now 1 ; : : : ; B are iid draws from h( jxn ). A histogram of these values provides an estimate of h( jxn ). 12.5 Large Sample Properties of Bayes' Procedures. In the Bernoulli and Normal examples we saw that the posterior mean was close to the mle . This is true in greater generality. Theorem 12.5 Under appropriate regularity conditions, we have 2 b se b ) where bn is the that the posteriorqis approximately N (; and seb = 1= nI (bn ). Hence, n bn . Also, if C = (bn z=2 seb ; bn + z=2 seb ) is the asymptotic frequentist 1 condence interval, then Cn is also an approximate 1 Bayesian posterior interval: P( 2 C jX n ) ! 1 : mle There is also a Bayesian delta method. Let = g (). Then 2 jX n N (b; se e ) 12.6 Flat Priors, Improper Priors and \Noninformative" Priors. 213 where b = g (b) and se e = se jg 0 (b)j. 12.6 Flat Priors, Improper Priors and \Noninformative" Priors. A big question in Bayesian inference is: where do you get the prior f ()? One school of thought, called \subjectivism" says that the prior should reect our subjective opinion about before the data are collected. This may be possible in some cases but seems impractical in complicated problems especially if there are many parameters. An alternative is to try to dene some sort of \noninformative prior." An obvious candidate for a noninformative prior is to use a \at" prior f () / constant. In the Bernoulli example, taking f (p) = 1 leads to pjX n Beta(s + 1; n s + 1) as we saw earlier which seemed very reasonable. But unfettered use of at priors raises some questions. Consider the N (; 1) example. Suppose we adopt a at prior f () / c where c > 0 is a constant. Note R that f ()d = 1 so this is not a real probability density in the usual sense. We call such a prior an improper prior. Nonetheless, we can still carry out Bayes' theorem and compute the posterior density f () / Ln ()f () / Ln (). In the normal example, this gives jX n N (X; 2 =n) and the resulting point and interval estimators agree exactly with their frequentist counterparts. In general, improper priors are not a problem as long as the resulting posterior is a well dened probability distribution. Improper Priors. Flat Priors are Not Invariant. Go back to the Bernoulli example and consider using the at prior f (p) = 1. Recall that a at prior presumably represents our lack of information about p before the experiment. Now let = log(p=(1 p)). This is a transformation and we can compute the resulting distribution 214 12. Bayesian Inference for . It turns out that f ( ) = e : (1 + e )2 But one could argue that if we are ignorant about p then we are also ignorant about so shouldn't we use a at prior for ? This contradicts the prior f ( ) for that is implied by using a at prior for p. In short, the notion of a at prior is not well-dened because a at prior on a parameter does not imply a at prior on a transformed version of the parameter. Flat priors are not transformation invariant. Jereys came up with a \rule" for creating priors. The rule is: take f () / I ()1=2 where I () is the Fisher information function. This rule turns out to be transformation invariant. There are various reasons for thinking that this prior might be a useful prior but we will not go into details here. Example 12.6 Consider the Bernoulli (p). Recall that Jeffreys' Prior. I (p) = 1 p(1 p) : Jerey's rule says to use the prior f (p) / p I (p) = p 1=2 (1 p) 1=2 : This is a Beta (1/2,1/2) density. This is very close to a uniform density. In a multiparameter problem, the Jereys' prior is dened to p be f () / detI () where det(A) denotes the determinant of a matrix A. 12.7 Multiparameter Problems 12.7 Multiparameter Problems 215 In principle, multiparameter problems are handled the same way. Suppose that = (1 ; : : : ; p ). The posterior density is still given by p(jxn ) / Ln ()f (): The question now arises of how to extract inferences about one parameter. The key is nd the marginal posterior density for the parameter of interest. Suppose we want to make inferences about 1 . The marginal posterior for 1 is f (1 j xn ) = Z Z f (1; ; pjxn )d2 : : : dp: In practice, it might not be feasible to do this integral. Simulation can help. Draw randomly from the posterior: 1 ; : : : ; B f (jxn) where the superscripts index the dierent draws. Each j is a vector j = (1j ; : : : ; pj ). Now collect together the rst component of each draw: 11 ; : : : ; 1B : These are a sample from f (1 jxn ) and we have avoided doing any integrals. Example 12.7 (Comparing two binomials.) Suppose we have n1 control patients and n2 treatment patients and that X1 control patients survive while X2 treatment patients survive. We want to estimate = g (p1 ; p2 ) = p2 p1 . Then, X1 Binomial(n1 ; p1 ) and X2 Binomial(n2 ; p2 ): Suppose we take f (p1 ; p2 ) = 1. The posterior is f (p1 ; p2 jx1 ; x2 ) / px1 1 (1 p1 )n1 x1 px2 (1 2 p2 )n2 x2 : Notice that (p1 ; p2 ) live on a rectangle (a square, actually) and that f (p1 ; p2 jx1 ; x2 ) = f (p1 jx1 )f (p2 jx2 ) 216 12. Bayesian Inference where f (p1 jx1 ) / px1 1 (1 p1 )n1 x1 and f (p2 jx2 ) / px2 2 (1 p2 )n2 x2 which implies that p1 and p2 are independent under the posterior. Also, p1 jx1 Beta(x1 + 1; n1 x1 + 1) and p2 jx2 Beta(x2 +1; n2 x2 +1). If we simulate P1;1 ; : : : ; P1;B Beta(x1 + 1; n1 x1 +1) and P2;1 ; : : : ; P2;B Beta(x2 +1; n2 x2 +1) then b = P2;b P1;b , b = 1; : : : ; B , is a sample from f ( jx1 ; x2 ). 12.8 Strengths and Weaknesses of Bayesian Inference Bayesian inference is appealing when prior information is available since Bayes' theorem is a natural way to combine prior information with data. Some people nd Bayesian inference psychologically appealing because it allows us to make probability statements about parameters. In contrast, frequentist inference provides condence sets Cn which trap the parameter 95 per cent of the time, but we cannot say that P( 2 CnjX n ) is .95. In the frequentist approach we can make probability statements about Cn not . However, psychological appeal is not really an argument for using one type of inference over another. In parametric models, with large samples, Bayesian and frequentist methods give approximately the same inferences. In general, they need not agree. Consider the following example. Example 12.8 Let X N (; 1) and suppose we use the prior N (0; 2 ). From (12.5), the posterior is x 1 jx N ; = N (cx; c) 1 1 + 2 1 + 12 where c = 2 =( 2 + 1). A 1 per cent posterior interval is C = (a; b) where p p a = cx cz=2 and b = cx + cz=2 : 12.9 Appendix 217 Thus, P( 2 C jX ) = 1 . We can now ask, from a frequntist perspective, what is the coverage of C , that is, how often will this interval contain the true value? The answer is P (a < < b) = = = p p p P (cX z=2 c < < cX + z=2 c) + z=2 c z=2 c c <X < P c c (1 c) z=2 c (1 c) + z=2 P <Z< c c p c p pc p p (1 c) + z=2 c (1 c) z=2 c = c c where Z N (0; 1). Figure 12.1 shows the coverage as a function of for = 1 and = :05. Unless the true value of is close to 0, the coverage can be very small. Thus, upon repeated use, the Bayesian 95 per cent interval might contain the true value with frequency near 0! In contrast, a condence interval has coverage 95 per cent coverage no matter what the true value of is. What should we conclude from all this? The important thing is to understand that frequentist and Bayesian methods are answering dierent questions. To combine prior beliefs with data in a principled way, use bayesian inference. To construct procedures with guaranteed long run performance, such as condence intervals, use frequentist methods. It is worth remarking that it is possible to develop nonparametric Bayesian methods similar to plug-in estimation and the bootstrap. be forewarned, however, that the frequency properties of nonparametric Bayesian methods can sometimes be quite poor. 12.9 Appendix Proof of Theorem 12.5. It can be shown that the eect of the prior diminishes as n increases so that f (jX n) / Ln()f () Ln(). Hence, 12. Bayesian Inference 0.4 0.0 0.2 coverage 0.6 0.8 1.0 218 0 1 2 3 4 5 FIGURE 12.1. Frequentist coverage of 95 per cent Bayesian posterior interval as a function of the true value . The dotted line marks the 95 per cent level. 12.10 Bibliographic Remarks 219 log f (jX n) `(). Now, `() `(b) + ( b)`0 (b) + [( b)2 =2]`00 (b) = `(b) + [( b)2 =2]`00 (b) since `0 (b) = 0. Exponentiating, we get approximately that f (jX n) / exp ( 1 ( b)2 2 n2 ) where n2 = 1=`00 (bn ). So the posterior of is approximately Normal with mean b and variance n2 . Let `i = log f (Xi j), then n 2 = = n = X `00 (bn ) = 1 X n nI (bn ) i i `00i (bn ) `00i (bn ) nE h i `00i (bn ) and hence n se(b). 12.10 Bibliographic Remarks Some references on Bayesian inference include Carlin and Louis (1996), Gelman, Carlin, Stern and Rubin (1995), Lee (1997), Robert (1994) and Schervish (1995). See Cox (1997), Diaconis and Freedman (1996), Freedman (2001), Barron, Schervish and Wasserman (1999), Ghosal, Ghosh and van der Vaart (2001), Shen and Wasserman (2001) and Zhao (2001) for discussions of some of the technicalities of nonparametric Bayesian inference. 12.11 Exercises 1. Verify (12.5). 2. Let X1 ; :::; Xn Normal(; 1): (a) Simulate a data set (using = 5) consisting of n=100 observations. (b) Take f () = 1 and nd the posterior density. Plot the density. 220 12. Bayesian Inference (c) Simulate 1000 draws from the posterior. Plot a histogram of the simulated values and compare the histogram to the answer in (b). (d) Let = e . Find the posterior density for analytically and by simulation. (e) Find a 95 per cent posterior interval for . (f) Find a 95 per cent condence interval for . 3. Let X1 ; :::; Xn Uniform(0; ): Let f () posterior density. / 1=. Find the 4. Suppose that 50 people are given a placebo and 50 are given a new treatment. 30 placebo patients show improvement while 40 treated patients show improvement. Let = p2 p1 where p2 is the probability of improving under treatment and p1 is the probability of improving under placebo. (a) Find the mle of . Find the standard error and 90 per cent condence interval using the delta method. (b) Find the standard error and 90 per cent condence interval using the parametric bootstrap. (c) Use the prior f (p1 ; p2 ) = 1. Use simulation to nd the posterior mean and posterior 90 per cent interval for . (d) Let = log p1 1 p1 1 p2 p2 be the log-odds ratio. Note that = 0 if p1 = p2 . Find the mle of . Use the delta method to nd a 90 per cent condence interval for . (e) Use simulation to nd the posterior mean and posterior 90 per cent interval for . 12.11 Exercises 221 5. Consider the Bernoulli(p) observations 0101000000 Plot the posterior for p using these priors: Beta(1/2,1/2), Beta(1,1), Beta(10,10), Beta(100,100). 6. Let X1 ; : : : ; Xn Poisson(). (a) Let Gamma(; ) be the prior. Show that the posterior is also a Gamma. Find the posterior mean. (b) Find the Jereys' prior. Find the posterior. 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D qQn l qQn ¯ !# = v yF® ® ¯# Vy;h L K v= ¿ ¯V# ce Mf ¿ K'0 , Q $ K¯ # 0 Q K & ¯#¬c* Vf Q ¯ & # Q ¯ & N Q Q Q Q , °l nf ±3ium s l qSs8²³s t qQn)(- K qQn l @pfpo¢o f t fnpo¢qSs l o iut npq m* f v= K ¯]/ & & B f t - µ dgf m Q qQn ºalt · f)- v= K ¿¯ d l n½nps lºSº f t½t qQn)( f * fD+o v= K ¿¯ » i µf kiutl nps lºSºt f · q ium q m oSdgfFD ltEl s´fpof t nD l* f m f lt]K1#À0 »%bed¸² n)s lm 5 Dgf i D º f D t f k f t K ¿ o i K »bed¸qQn3q ºSº ²npo tEl ofn oSd l o ium f 3 m ²¸s;@pf t & np²³s8s lt q fn º q ( f8s l qSs8²¸s t qQn)( lt f qSsFg D f tSk f * o » +/i µ *ium npq ± f t oSdgf ( »"8 iut q ºSº ² npo tEl orq ium - º fpo ² n§o l ( f %K ¯ #c0 »°bedgf m °l 5 fn t qQn) ® K ¿`¯V# y v=%K® K ¿`¯EK K´# y K 0 $ K ¯ K K# 0 Q Q v= lm+± ® K & ¯#zy v=%K® K & ¯'KK´# Q ¯ N Q Q & - - ® K & ¯ ® K ¿l¯ µ d¸q * d8np² ·· fnpoHn°oSd l o K 8 Q @ i t u ¿ qQn l pfpo¢o f t fnpo¢qSs l o iut »bed¸qQn3s8q · d¸o nffps q m ¢o ²¸qSo¢q f º t f l n ium+l º f ;²³o§oHd¸qQn lm npµf t ± f gf m+± n ium oHdgf d i q f ik t q iut » bedgf l ± o lm o l· f ik ² npq m¸· s l qSs8²³s t qQn s n qQnoSd l o°Sq o ±¸i fn m i o t f ¹;²¸q t f ± fn +qSofqSoSn t[i º fpÃ D )(- D * 4B 65 D * D @ )- B @ [email protected] 3 !" 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B '0ÁFu=Á¯ µ dgf t f§oHdgf§q m s8²¸sÂqQn i f t/lºSº fnporqSs l o iut n » R 43 TVUOW+XLY Z[Y \]X3^_]` m fnporqSs l o iut oHd l o3s8q m qSs8qufn oHdgf s l q s8²¸s t qQn §qQn lºSº f ± ½ l py¦r'§r ¤¥ » iut s lºSº Qq n s8q m qSs l q k v P"® = ¯V#¬#q ! v=P"® ¯ '0ÁFu b¯ ? ? µ dgf t f§oHdgf§q m s8²¸sÂqQn i f t/lºSº fnporqSs l o iut n » )( * 8 65 - 3 B DFEHGE oT LFW hW¡ ORQ T¡UK W 0z Y$ s]u°`]c ¢P [{ /_]_c|mU]`/_]^#x{ F P( GU¯V# HGV( ¯ y¯ S GU¯ F # 5 F S GV( ¯ y¯ F HGV( ¯ P¯EKb '0Á#u ¯ }_ HGU¯# 5 HG0®/¯EKb # 5 SGV( ¯ P¯EKbrh '§P0§¥ P_[ ¦¤0M']qR£F u / F _bVydVy;hY] q, _`c* /] % « p" % !' 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The covariate is also called a predictor variable or a feature. Later, we will generalize and allow for more than one covariate. The The term \regression" is due to data are of the form Sir Francis Galton (Y1 ; X1 ); : : : ; (Yn; Xn): (1822-1911) who One way to summarize the relationship between X and Y is noticed that tall and short men tend through the regression function to have sons with Z r(x) = E (Y jX = x) = y f (y jx)dy: (14.1) heights closer to the mean. He called this Most of this chapter is concerned with estimating the regression \regression towards the mean." function. 14.1 Simple Linear Regression The simplest version of regression is when Xi is simple (a scalar not a vector) and r(x) is assumed to be linear: r(x) = 0 + 1 x: 246 14. Linear Regression This model is called the the simple linear regression model. Let i = Yi (0 + 1 Xi ). Then, E (i jXi) = = = = (0 + 1 Xi )jXi) = E (Yi jXi ) r(Xi) (0 + 1 Xi ) (0 + 1 Xi ) (0 + 1 Xi ) 0: E (Yi (0 + 1 Xi ) Let 2 (x) = V (i jX = x). We will make the further simplifying assumption that 2 (x) = 2 does not depend on x. We can thus write the linear regression model as follows. Denition 14.1 The Linear Regresion Model Yi = 0 + 1 Xi + i (14.2) where E (i jXi) = 0 and V (i jXi ) = 2 . Example 14.2 Figure 14.1 shows a plot of Log surface temperature (Y) versus Log light intensity (X) for some nearby stars. Also on the plot is an estimated linear regression line which will be explained shortly. The unknown parameters in the model are the intercept 0 and the slope 1 and the variance 2 . Let b0 and b1 denote estimates of 0 and 1 . The tted line is dened to be rb(x) = b0 + b1 x: (14.3) The predicted values or tted values are Ybi = rb(Xi ) and the residuals are dened to be b b b bi = Yi Yi = Yi 0 + 1 Xi : (14.4) The residual sums of squares or rss is dened by n X rss = b2i : i=1 247 4.4 4.3 4.2 4.1 4.0 Log surface temperature (Y) 4.5 4.6 14.1 Simple Linear Regression 4.0 4.5 5.0 Log light intensity (X) FIGURE 14.1. Data on stars in nearby stars. 5.5 248 14. Linear Regression The quantity rss measures how well the tted line ts the data. Denition 14.3 The least squares estimates are the valP ues b0 and b1 that minimize rss = ni=1 b2i . Theorem 14.4 The least squares estimates are given by Pn (Xi X n )(Yi Y n ) i=1P b 1 = ; (14.5) n 2 i=1 (Xi X n ) b0 = Y n b1 X n : (14.6) An unbiased estimate of 2 is b = 2 1 n 2 X n i=1 b2i : (14.7) Example 14.5 Consider the star data from Example 14.2. The least squares estimates are b0 = 3:58 and b1 = 0:166. The tted line rb(x) = 3:58 + 0:166x is shown in Figure 14.1. Example 14.6 (The 2001 Presidential Election.) Figure 14.2 shows the plot of votes for Buchanan (Y) versus votes for Bush (X) in Florida. The least squares estimates (omitting Palm Beach County) and the standard errors are b0 = 66:0991 seb (b0 ) = 17:2926 b1 = 00:0035 seb (b1 ) = 0:0002: The tted line is Buchanon = 66:0991 + :0035 Bush: (We will see later how the standard errors were computed.) Figure 14.2 also shows the residuals. The inferences from linear 14.2 Least Squares and Maximum Likelihood 249 regression are most accurate when the residuals behave like random normal numbers. Based on the residual plot, this is not the case in this example. If we repeat the analysis replacing votes with log(votes) we get b0 = 2:3298 seb (b0 ) = :3529 b1 = 0:730300 seb (b1 ) = 0:0358: This gives the t log(Buchanon) = 2:3298 + :7303 log(Bush): The residuals look much healthier. Later, we shall address two interesting questions: (1) how do we see if Palm Beach County has a statistically plausible outcome? (2) how do we do this problem nonparametrically? 14.2 is, Least Squares and Maximum Likelihood Suppose we add the assumption that i jXi N (0; ), that 2 Yi jXi N (i ; i2 ) where i = 0 + 1 Xi . The likelihood function is n Y i=1 n Y f (Xi ; Yi ) = i=1 n Y = where L1 = i=1 i=1 fX (Xi ) L 2 fX (Xi ) L L = Qn fX (Xi )fY jX (YijXi ) 1 n Y i=1 fY jX (Yi jXi ) 2 and = n Y i=1 fY jX (Yi jXi ): (14.8) Linear Regression • •• • • • • • • • • • • • • • • • • • ••••••••• •• • • 0 50000 200 • • • • • ••••• • ••••••••••• •••• ••• • •••• • •• • • • • • • •• • 150000 0 Residuals 1000 3000 • 250000 0 50000 150000 1.0 • 0.0 • -1.0 Residuals 8 7 6 5 4 3 •• 2 log Buchanon • • •• • • • • • • • • ••• ••• •• • • • • • • • •• • • • • ••• • • • •• • • • •••• •• • •• •• • 8 9 10 250000 Bush • 7 •• • Bush • • -400 14. 0 Buchanon 250 11 12 log Bush FIGURE 14.2. Voting Data for Election 2000. 7 • • • • • • • • • • •• •• • • • • • • • • •• • •• • • • • • • • • • • • •• ••• • • • •• • • • • • • • •• • • • • • • • 8 9 10 log Bush 11 12 14.3 Properties of the Least Squares Estimators 251 The term L1 does not involve the parameters 0 and 1 . We shall focus on the second term L2 which is called the conditional likelihood, given by ( ) n Y X 1 L2 L(0; 1; ) = fY jX (YijXi) / n exp 22 (Yi i)2 : i=1 i The conditional log-likelihood is `(0 ; 1 ; ) = n log n 1 X 2 2 i=1 Yi 2 (0 + 1 Xi ) : (14.9) To nd the mle of (0 ; 1 ) we maximize `(0 ; 1 ; ). From (14.9) we see that maximizing the likelihood is the same as minimizing 2 Pn the rss i=1 Yi (0 + 1 Xi ) . Therefore, we have shown the following. Theorem 14.7 Under the assumption of Normality, the least squares estimator is also the maximum likelihood estimator. We can also maximize `(0 ; 1 ; ) over yielding the mle 1X 2 b2 = bi : (14.10) n i This estimator is similar to, but not identical to, the unbiased estimator. Common practice is to use the unbiased estimator (14.7). 14.3 Properties of the Least Squares Estimators We now record the standard errors and limiting distribution of the least squares estimator. In regression problems, we usually focus on the properties of the estimators conditional on X n = (X1 ; : : : ; Xn ). Thus, we state the means and variances as conditional means and variances. 252 14. Linear Regression Theorem 14.8 Let bT = (b0 ; b1 )T denote the least squares estimators. Then, 0 n b E ( jX ) = 1 ! Pn 1 2 2 X X n n i=1 i V (bjX n ) = (14.11) n s2X Xn 1 P where s2X = n 1 ni=1 (Xi X n )2 . The estimated standard errors of b0 and b1 are obtained by taking the square roots of the corresponding diagonal terms of V (bjX n ) and inserting the estimate b for . Thus, r Pn 2 b i=1 Xi p b (b0 ) = se (14.12) n sX n b p : b (b1 ) = se sX n (14.13) b (b0 jX n ) and seb (b1 jX n) but We should really write these as se b (b0 ) and seb (b1 ). we will use the shorter notation se Theorem 14.9 Under appropriate conditions we have: 1. (Consistency): b0 ! P 0 and b1 ! . P 1 2. (Asymptotic Normality): b0 0 b (b0 ) se 3. Approximate 1 N (0; 1) and b1 1 b (b1 ) se N (0; 1): condence intervals for 0 and 1 are b0 z=2 seb (b0 ) and b1 z=2 seb (b1 ): (14.14) The Wald test statistic for testing H0 : 1 = 0 versus H1 : 1 6= 0 is: reject H0 if W > z=2 where W = b1 =seb (b1 ). 14.4 Prediction 253 Example 14.10 For the election data, on the log scale, a 95 per cent condence interval is :7303 2(:0358) = (:66; :80). The fact that the interval excludes 0 The Wald statistics for testing H0 : 1 = 0 versus H1 : 1 6= 0 is W = j:7303 0j=:0358 = 20:40 with a p-value of P(jZ j > 20:40) 0. This is strong evidence that that the true slope is not 0. 14.4 Prediction Suppose we have estimated a regression model rb(x) = b0 + b1 from data (X1 ; Y1 ); : : : ; (Xn; Yn ). We observe the value X = x of the covariate for a new subject and we want to predict their outcome Y . An estimate of Y is Yb = b0 + b1 X : (14.15) Using the formula for the variance of the sum of two random variables, V (Yb ) = V (b0 + b1 x ) = V (b0 ) + x2 V (b1 ) + 2x Cov(b0 ; b1 ): Theorem 14.8 gives the formulas for all the terms in this equab (Yb ) is the square root of tion. The estimated standard error se this variance, with b2 in place of 2 . However, the condence interval for Y is not of the usual form Yb z=2 . The appendix explains why. The correct form of the condence interval is given in the following Theorem. We call the interval a prediction interval. 254 14. Linear Regression Theorem 14.11 (Prediction Interval) Let bn2 = seb 2 (Yb ) + b2 Pn (Xi X )2 2 i =1 +1 : = b P n i (Xi X )2 An approximate 1 (14.16) predicition interval for Y is Yb z=2 n: (14.17) Example 14.12 (Election Data Revisited.) On the log-scale, our linear regression gives the following prediction equation: log (Buchanon) = 2:3298 + :7303log (Bush). In Palm Beach, Bush had 152954 votes and Buchanan had 3467 votes. On the log scale this is 11.93789 and 8.151045. How likely is this outcome, assuming our regression model is appropriate? Our prediction for log Buchanan votes -2.3298 + .7303 (11.93789)=6.388441. Now 8.151045 is bigger than 6.388441 but is is \signicantly" bigger? Let us compute a condence interval. We nd that bn = :093775 and the approximate 95 per cent condence interval is (6.200,6.578) which clearly excludes 8.151. Indeed, 8.151 is nearly 20 standard errors from Yb . Going back to the vote scale by exponentiating, the condence interval is (493,717) compared to the actual number of votes which is 3467. 14.5 Multiple Regression Now suppose we have k covariates X1 ; : : : ; Xk . The data are of the form (Y1 ; X1 ); : : : ; (Yi; Xi ); : : : ; (Yn; Xn ) where Xi = (Xi1 ; : : : ; Xik ): 14.5 Multiple Regression 255 Here, Xi is the vector of k covariate values for the ith observation. The linear regression model is Yi = k X j =1 j Xij + i (14.18) for i = 1; : : : ; n, where E (i jX1i ; : : : ; Xki) = 0. Usually we want to include an intercept in the model which we can do by setting Xi1 = 1 for i = 1; : : : ; n. At this point it will be more convenient to express the model in matrix notation. The outcomes will be denoted by 0 1 B Y =B B @ Y1 Y2 C C .. C . A Yn and the covariates will be denoted by 0 1 X11 X12 : : : X1k B X21 X22 : : : X2k C C: X =B B .. C . . . . . . @ . . . . A Xn1 Xn2 : : : Xnk Each row is one observation; the columns correspond to the k covariates. Thus, X is a (n k) matrix. Let 0 1 0 1 1 1 k n C B C =B @ ... A and = @ ... A : Then we can write (14.18) as Y = X + : (14.19) Theorem 14.13 Assuming that the (k k) matrix X T X is invertible, the least squares estimate is 256 14. Linear Regression b = (X T X ) 1 X T Y: (14.20) The estimated regression function is rb(x) = k X j =1 bj xj : (14.21) The variance-covariance matrix of b is j V (b X n ) = 2 (X T X ) 1 : Under appropriate conditions, b N (; 2 (X T X ) 1 ): An unbiased estimate of 2 is b = 2 1 n k X n i=1 b2i where b = X b Y is the vector of residuals. An approximate 1 condence interval for j is bj z=2 seb (bj ) (14.22) b 2 (bj ) is the j th diagonal element of the matrix b2 (X T X ) 1 . where se Example 14.14 Crime data on 47 states in 1960 can be obtained at http://lib.stat.cmu.edu/DASL/Stories/USCrime.html. If we t a linear regression of crime rate on 10 variables we get the following: 14.6 Model Selection 257 Covariate Least Estimated t value p-value Squares Standard Estimate Error (Intercept) -589.39 167.59 -3.51 0.001 ** Age 1.04 0.45 2.33 0.025 * Southern State 11.29 13.24 0.85 0.399 Education 1.18 0.68 1.7 0.093 Expenditures 0.96 0.25 3.86 0.000 *** Labor 0.11 0.15 0.69 0.493 Number of Males 0.30 0.22 1.36 0.181 Population 0.09 0.14 0.65 0.518 Unemployment (14-24) -0.68 0.48 -1.4 0.165 Unemployment (25-39) 2.15 0.95 2.26 0.030 * Wealth -0.08 0.09 -0.91 0.367 This table is typical of the output of a multiple regression program. The \t-value" is the Wald test statistic for testing H0 : j = 0 versus H1 : j 6= 0. The asterisks denote \degree of signicance" with more asterisks being signicant at a smaller level. The example raises several important questions. In particular: (1) should we eliminate some variables from this model? (2) should we interpret this relationships as causal? For example, should we conclude that low crime prevention expenditures cause high crime rates? We will address question (1) in the next section. We will not address question (2) until a later Chapter. 14.6 Model Selection Example 14.14 illustrates a problem that often arises in multiple regression. We may have data on many covariates but we may not want to include all of them in the model. A smaller model with fewer covariates has two advantages: it might give better predictions than a big model and it is more parsimonious (simpler). Generally, as you add more variables to a regression, 258 14. Linear Regression the bias of the predictions decreases and the variance increases. Too few covariates yields high bias; too many covariates yields high variance. Good predictions result from achieving a good balance between bias and variance. In model selection there are two problems: (i) assigning a \score" to each model which measures, in some sense, how good the model is and (ii) searching through all the models to nd the model with the best score. Let us rst discuss the problem of scoring models. Let S f1; : : : ; kg and let XS = fXj : j 2 S g denote a subset of the covariates. Let S denote the coeÆcients of the corresponding set of covariates and let bS denote the least squares estimate of S . Also, let XS denote the X matrix for this subset of covariates and dene rbS (x) to be the estimated regression function from (14.21). The predicted valus from model S are denoted by Ybi (S ) = rbS (Xi ). The prediction risk is dened to be R(S ) = n X i=1 E (Ybi (S ) Yi )2 (14.23) where Yi denotes the value of a future observation of Yi at covariate value Xi . Our goal is to choose S to make R(S ) small. The training error is dened to be Rbtr (S ) = n X i=1 (Ybi (S ) Yi )2 : This estimate is very biased and under-estimates R(S ). Theorem 14.15 The training error is a downward biased estimate of the prediction risk: btr (S )) < R(S ): E (R In fact, bias (Rbtr (S )) = E (Rbtr (S )) R(S ) = 2 X i=1 Cov(Ybi; Yi ): (14.24) 14.6 Model Selection 259 The reason for the bias is that the data are being used twice: to estimate the parameters and to estimate the risk. When we t a complex model with many parameters, the covariance Cov(Ybi ; Yi) will be large and the bias of the training error gets worse. In summary, the training error is a poor estimate of risk. Here are some better estimates. Mallow's Cp statistic is dened by Rb(S ) = Rbtr (S ) + 2jS jb2 (14.25) where jS j denotes the number of terms in S and b2 is the estimate of 2 obtained from the full model (with all covariates in the model). This is simply the training error plus a bias correction. This estimate is named in honor of Colin Mallows who invented it. The rst term in (14.25) measures the t of the model while the second measure the complexity of the model. Think of the Cp statistic as: lack of t + complexity penalty: Thus, nding a good model involves trading o t and complexity. A related method for estimating risk is AIC (Akaike Information Criterion). The idea is to choose S to maximize Some texts use a slightly dierent `S jS j (14.26) denition of AIC which involves where `S is the log-likelihood of the model evaluated at the multiplying the mle . This can be thought of \goodness of t" minus \com- denition here by plexity." In linear regression with Normal errors, maximizing 2 or -2. This has no eect on which AIC is equivalent to minimizing Mallow's Cp ; see exercise 8. Yet another method for estimating risk is leave-one-out model is selected. cross-validation. In this case, the risk estimator is RbCV (S ) = n X ( Yi i=1 Yb(i) )2 (14.27) 260 14. Linear Regression where Yb(i) ) is the prediction for Yi obtained by tting the model with Yi omitted. It can be shown that ! n X bi (S ) 2 Y Y i (14.28) RbCV (S ) = 1 Uii (S ) i=1 where Uii (S ) is the ith diagonal element of the matrix U (S ) = XS (XST XS ) 1 XST : (14.29) Thus, one need not actually drop each observation and re-t the model. A generalization is k-fold cross-validation. Here we divide the data into k groups; often people take k = 10. We omit one group of data and t the models to the remaining data. We use the tted model to predict the data in the group that P was omitted. We then estimate the risk by i (Yi Ybi )2 where the sum is over the the data points in the omitted group. This process is repeated for each of the k groups and the resulting risk estimates are averaged. For linear regression, Mallows Cp and cross-validation often yield essentially the same results so one might as well use Mallows' method. In some of the more complex problems we will discuss later, cross-validation will be more useful. Another scoring method is BIC (Bayesian information criterion). Here we choose a model to maximize BIC(S ) = rss (S ) + 2jS jb2: (14.30) The BIC score has a Bayesian interpretation. Let S = fS1 ; : : : ; Sm g denote a set of models. Suppose we assign the prior P(Sj ) = 1=m over the models. Also, assume we put a smooth prior on the parameters within each model. It can be shown that the posterior probability for a model is approximately, j P(Sj data) BIC (Sj ) Pe eBIC S : r ( r) 14.6 Model Selection 261 Hence, choosing the model with highest BIC is like choosing the model with highest posterior probability. The BIC score also has an information-theoretic interpretation in terms of something called minimum description length. The BIC score is identical to Mallows Cp except that it puts a more severe penalty for complexity. It thus leads one to choose a smaller model than the other methods. Now let us turn to the problem of model search. If there are k covariates then there are 2k possible models. We need to search through all these models, assign a score to each one, and choose the model with the best score. If k is not too large we can do a complete search over all the models. When k is large, this is infeasible. In that case we need to search over a subset of all the models. Two common methods are forward and backward stepwise regression. In forward stepwise regression, we start with no covariates in the model. We then add the one variable that leads to the best score. We continue adding variables one at a time until the score does not improve. Backwards stepwise regression is the same except that we start with the biggest model and drop one variable at a time. Both are greedy searches; nether is guaranteed to nd the model with the best score. Another popular method is to do random searching through the set of all models. However, there is no reason to expect this to be superior to a deterministic search. Example 14.16 We apply backwards stepwise regression to the crime data using AIC. The following was obtained from the program R. This program uses minus our version of AIC. Hence, we are seeking the smallest possible AIC. This is the same is minimizing Mallows Cp . 262 14. Linear Regression The full model (which includes all covariates) has AIC= 310.37. The AIC scores, in ascending order, for deleting one variable are as follows: variable Pop Labor South Wealth Males U1 Educ. U2 Age Expend AIC 308 309 309 309 310 310 312 314 315 324 For example, if we dropped Pop from the model and kept the other terms, then the AIC score would be 308. Based on this information we drop \population" from the model and the current AIC score is 308. Now we consider dropping a variable from the current model. The AIC scores are: variable South Labor Wealth Males U1 Education U2 Age Expend AIC 308 308 308 309 309 310 313 313 329 We then drop \Southern" from the model. This process is continued until there is no gain in AIC by dropping any variables. In the end, we are left with the following model: Crime = 1:2 Age + :75 Education + :87 Expenditure + :34 Males Warning! This does not yet address the question of which variables are causes of crime. 14.7 The Lasso There is an easier model search method although it addresses a slightly dierent question. The method, due to Tibshirani, is called the Lasso. In this section we assume that the covariates have all been rescaled to have the same variance. This puts each covariate on the same scale. Consider estimating = (1 ; : : : ; k ) by minimizing the loss function n X i=1 (Yi Ybi )2 + k X j =1 jj j (14.31) :86 U1 + 2:31 U2: 14.8 Technical Appendix 263 where > 0. The idea is to minimize the sums of squares but we include a penalty that gets large if any of the j0 s are large. The solution b = (b1 ; : : : ; bk ) can be found numerically and will depend on the choice of . It can be shown that some of the bj 's will be 0. We interpret this is meaning that the j th is omitted from the model. Hence, we are doing estimation and model selection simultaneously. We need to choose a value of . We can do this by estimating the prediction risk R() as a function of and choosing to minimze the estimated risk. For example, we can estimate the risk using leave-one-out crossvalidation. Example 14.17 Returning to the crime data, Figure 14.3 shows the results of the lasso. The rst plot shows the leave-one-out cross-validation score as a function of . The minimum occurs at = :7. The second plot shows the estimated coeÆcients as a function of . You can see how some estimated parameters are zero until gets larger. At = :7, all the bj are non-zero so the Lasso chooses the full model. 14.8 Technical Appendix Why is the predicition interval of a dierent form than the other condence intervals we have seen? The reason is that the quantity we want to estimate, Y , is not a xed parameter, it is a random variable. To understand this point better, let = 0 + 1 X and let b = b0 + b1 X . Thus, Yb = b while Y = + . Now, b N (; se 2 ) where se 2 = V (b) = V (b0 + b1 x ): q Note that V (b) is the same as V (Yb ). Now, b 2 V ar(b) is a 95 per cent condence interval for = 0 + 1 x using the usual Linear Regression 800 1200 14. cross-validation score 264 0.2 0.4 0.6 0.8 0.6 0.8 −10 b() 10 30 0.2 0.4 FIGURE 14.3. The Lasso applied to the crime data. 14.8 Technical Appendix 265 argument for a condence interval. It is not a valid condence intervalqfor Y . To see why, let's compute the probability that q Yb 2 V (Yb ) contains Y . Let s = V ar(Yb ). Then, ! b Y Y P(Yb 2s < Y < Yb + 2s) = P 2< <2 s ! b <2 = P 2< s ! b = P 2< <2 s s 2 P 2 < N (0; 1) N 0; s2 < 2 2 = P 2 < N 0; 1 + 2 < 2 s 6= :95: The problem is that the quantity of interest Y is equal to a parameter plus a random variable. We can x this by dening n = V (Yb ) + 2 = 2 P (xi x )2 i P + 1 2: n i (xi x)2 In practice, we substitute b for and we denote the resulting quantity by bn. Now consider the interval Yb 2bn. Then, ! b Y Y <2 P(Yb 2bn < Y < Yb + 2bn ) = P 2< bn ! b = P 2< <2 bn N (0; s2 + 2 ) P 2< <2 bn N (0; s2 + 2 ) P 2< <2 n 266 14. Linear Regression = P ( 2 < N (0; 1) < 2) = :95: Of course, a 1 14.9 interval is given by Yb z=2 bn . Exercises 1. Prove Theorem 14.4. 2. Prove the formulas for the standard errors in Theorem 14.8. You should regard the Xi 's as xed constants. 3. Consider the regression through the origin model: Yi = Xi + : Find the least squares estimate for . Find the standard error of the estimate. Find conditions that guarantee that the estimate is consistent. 4. Prove equation (14.24). 5. In the simple linear regression model, construct a Wald test for H0 : 1 = 170 versus H1 : 1 6= 170 . 6. Get the passenger car mileage data from http://lib.stat.cmu.edu/DASL/Datales/carmpgdat.html (a) Fit a simple linear regression model to predict MPG (miles per gallon) from HP (horsepower). Summarize your analysis including a plot of the data with the tted line. (b) Repeat the analysis but use log(MPG) as the response. Compare the analyses. 7. Get the passenger car mileage data from http://lib.stat.cmu.edu/DASL/Datales/carmpgdat.html (a) Fit a multiple linear regression model to predict MPG (miles per gallon) from HP (horsepower). Summarize your analysis. 14.9 Exercises 267 (b) Use Mallow Cp to select a best sub-model. To search trhough the models try (i) all possible models, (ii) forward stepwise, (iii) backward stepwise. Summarize your ndings. (c) Repeat (b) but use BIC. Compare the results. (d) Now use the Lasso and compare the results. 8. Assume that the errors are Normal. Show that the model with highest AIC (equation (14.26)) is the model with the lowest Mallows Cp statistic. 9. In this question we will take a closer look at the AIC method. Let X1 ; : : : ; Xn be iid observations. Consider two models M0 and M1 . Under M0 the data are assumed to be N (0; 1) while under M1 the data are assumed to be N (; 1) for some unknown 2 R : M M : X1 ; : : : ; Xn 1 : X1 ; : : : ; Xn 0 N (0; 1) N (; 1); 2 R : This is just another way to view the hypothesis testing problem: H0 : = 0 versus H1 : 6= 0. Let `n () be the log-likelihood function. The AIC score for a model is the log-likelihood at the mle minus the number of parameters. (Some people multiply this score by 2 but that is irrelevant.) Thus, the AIC score for M0 is AIC0 = `n (0) and the AIC score for M1 is AIC1 = `n (b) 1. Suppose we choose the model with the highest AIC score. Let Jn denote the selected model: 0 > AIC1 Jn = 01 ifif AIC AIC1 > AIC0 : (a) Suppose that M0 is the true model, i.e. = 0. Find lim P (Jn = 0) : n!1 268 14. Linear Regression Now compute limn!1 P (Jn = 0) when 6= 0. (b) The fact that limn!1 P (Jn = 0) 6= 1 when = 0 is why some people say that AIC \overts." But this is not quite true as we shall now see. Let (x) denote a Normal density function with mean and variance 1. Dene 0 b fn (x) = 0 ((xx)) ifif JJn = n = 1: b p If = 0, show that D(0 ; fbn ) ! 0 as n ! 1 where Z f (x) D(f; g ) = f (x) log dx g (x) p is the Kullback-Leibler distance. Show also that D( ; fbn ) ! 0 if 6= 0. Hence, AIC consistently estimates the true density even if it \overshoots" the correct model. REMARK: If you are feeling ambitious, repeat this analysis for BIC which is the log-likelihood minus (p=2) log n where p is the number of parameters and n is sample size. This is page 269 Printer: Opaque this 15 Multivariate Models In this chapter we revisit the Multinomial model and the multivariate Normal, as we will need them in future chapters. Let us rst review some notation from linear algebra. ReP T call that if x and y are vectors then x y = j xj yj . If A is a matrix then det(A) denotes the determinant of A, AT denotes the transpose of A and A denotes the the inverse of A (if the inverse exists). The trace of a square matrix A { denoted by tr(A) { is the sum of its diagonal elements. The trace satises tr(AB ) = tr(BA) and tr(A) + tr(B ). Also, tr(a) = a if a is a scalar (i.e. a real number). A matrix is positive denite if xT x > 0 for all non-zero vectors x. If a matrix is symmetric and positive denite, there exists a matrix = { called the square root of { with the following properties: 1. = is symmetric; 2. = = = ; 3. = = = = = = I where = = ( = ) . 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 270 15.1 15. Multivariate Models Random Vectors Multivariate models involve0a random vector X of the form 1 X= B @ X1 ... C A: Xk The mean of a random vector X is dened by 0 1 0 1 1 E (X1 ) . C B ... C =B @ .. A = @ A: k E (Xk ) The covariance matrix is dened to be 2 V (X ) Cov(X ; X ) 6 Cov(X ; X ) V (X ) = V (X ) = 664 .. ... . Cov(Xk ; X ) Cov(Xk ; X ) 1 1 2 1 2 2 1 2 ... (15.1) 3 Cov(X1 ; Xk ) Cov(X2 ; Xk ) 7 7 ... V (Xk ) 7: 5 (15.2) This is also called the variance matrix or the variance-covariance matrix. a be a vector of length k and let X be a random vector of the same length with mean and variance . Then E (aT X ) = aT and V (aT X ) = aT a. If A is a matrix with k columns then E (AX ) = A and V (AX ) = AAT . Theorem 15.1 Let Now suppose we have a random sample of n vectors: 0 B B B @ X11 X21 ... Xk1 1 0 C C C; A B B B @ X12 X22 ... Xk2 1 C C C; A 0 :::; B B B @ X1n X2n ... Xkn The sample mean X is a vector dened by 0 X1 . X =B @ .. Xk 1 C A 1 C C C: A (15.3) 15.2 Estimating the Correlation 271 P n where X i = n j Xij . The sample variance matrix, also called the covariance matrix or the variance-covariance matrix, is 2 3 1 =1 S= where 6 6 6 4 s11 s12 s12 s22 s k s k s1k s2k skk ... ... 1 ... ... 2 7 7 7 5 (15.4) n X 1 sab = (Xaj X a )(Xbj X b): n 1j =1 It follows that E (X ) = . and E (S ) = . 15.2 Estimating the Correlation Consider n data points from a bivariate distribution: X11 X21 ; X12 X22 ;; X1n X2n : Recall that the correlation between X and X is E ((X )(X )) = : 1 1 1 1 2 2 2 2 (15.5) The sample correlation (the plug-in estimator) is b = Pn i=1 (X1i X 1 )(X2i s1 s2 X 2) : (15.6) We can construct a condence interval for by applying the delta method as usual. However, it turns out that we get a more accurate condence interval by rst constructing a condence interval for a function = f () and then applying the inverse function f . The method, due to Fisher, is as follows. Dene 1 f (r) = log(1 + r) log(1 r) 2 1 272 15. Multivariate Models akd let = f (). The inverse of r is g (z ) f 1 (z ) = e2z 1 : e2z + 1 Now do the following steps: Approximate Condence Interval for The Correlation 1. Compute b 1 = f (b) = 2 log(1 + b) log(1 b) : 2. Compute the approximate standard error of b which can be shown to be 1 : b (b) = p se n 3 3. An approximate 1 condence interval for = f () is (a; b) b pz=2 ; b + pz=2 : n 3 n 3 4. Apply the inverse transformation f (z) to get a condence interval for : a e 1; e b 1 : e a+1 e b+1 1 15.3 2 2 2 2 Multinomial Let us now review the Multinomial distribution. Consider drawing a ball from an urn which has balls with k dierent colors labeled color 1, color 2, ... , color k. Let p = (p ; : : : ; pk ) 1 15.3 Multinomial 273 P where pj 0 and kj pj = 1 and suppose that pj is the probability of drawing a ball of color j . Draw n times (independent draws with replacement) and let X = (X ; : : : ; Xk ) where Xj is Pk the number of times that color j appeared. Hence, n = j Xj . We say that X has a Multinomial (n,p) distribution. The probability function is =1 1 =1 n f (x; p) = px1 pxkk x1 : : : xk 1 where n n! = : x1 : : : xk x1 ! xk ! Theorem 15.2 Let X Multinomial(n; p). Then the marginal distribution of Xj is Xj Binomial(n; pj ). The mean and variance of X are 0 1 np1 B C E (X ) = @ ... A npk and V (X ) = 0 B B B @ np1 (1 p1 ) np1 p2 np1 p2 np2 (1 p2 ) .. . np1 pk .. . np2 pk .. . np1 pk np2 pk .. . npk (1 pk ) 1 C C C: A That Xj Binomial(n; pj ) follows easily. Hence, E (Xj ) = npj and V (Xj ) = npj (1 pj ). To compute Cov(Xi ; Xj ) we proceed as follows. Notice that Xi + Xj Binomial(n; pi + pj ) and so V (Xi + Xj ) = n(pi + pj )(1 pi pj ). On the other hand, Proof. V (Xi + Xj ) = V (Xi) + V (Xj ) + 2Cov(Xi; Xj ) = npi (1 pi) + npj (1 pj ) + 2Cov(Xi; Xj ): Equating this last expression with n(pi + pj )(1 pi pj ) implies that Cov(Xi; Xj ) = npi pj . 274 15. Theorem Multivariate Models 15.3 The maximum likelihood estimator of 0 pb = B @ 1 0 .. C . A pbk =B @ pb1 X1 n .. . Xk n p is 1 C A = Xn : The log-likelihood (ignoring the constant) is Proof. `(p) = k X j =1 Xj log pj : When we maximize `Pwe have to be careful since we must enforce the constraint that j pj = 1. We use the method of Lagrange multipliers and instead maximize A(p) = k X j =1 Now Xj log pj + X j pj 1 : @A(p) Xj = p + : @pj j P Setting @[email protected] p = 0 yields pbj = Xj =. Since j pbj = 1 we see that = n and hence pbj = Xj =n as claimed. Next we would like to know the variability of the mle . We can either compute the variance matrix of pb directly or we can approximate the variability of the mle by computing the Fisher information matrix. These two approaches give the same answer in this case. The direct approach is easy: V (pb) = V (X=n) = n V (X ) and so ( ) j 2 0 V (pb) = 1B B n B @ p1 (1 p1 ) p1 p2 p1 p2 p2 (1 p2 ) ... p1 p k ... p2 pk ... p1 pk p2 pk ... pk (1 pk ) 1 C C C: A 15.4 Multivariate Normal 15.4 275 Multivariate Normal Let us recall how the multivariate Normal distribution is dened. To begin, let 0 1 Z= Z1 ... B @ Zk C A where Z ; : : : ; Zk N (0; 1) are independent. The density of Z is 1 ( k X ) 2 1 z = 1 exp 2j j (2)k= 1 zT z : 2 (15.7) The variance matrix of Z is the identity matrix I . We write Z N (0; I ) where it is understood that 0 denotes a vector of k zeroes. We say that Z has a standard multivariate Normal distribution. More generally, a vector X has a multivariate Normal distribution, denoted by X N (; ), if its density is 1 exp f (z ) = (2)k= 2 2 =1 1 f (x; ; ) = (2)k= det() = exp 1 (x )T (x ) 2 (15.8) where det() denotes the determinant of a matrix, is a vector of length k and is a k k symmetric, positive denite matrix. Then E (X ) = and V (X ) = . Setting = 0 and = I gives back the standard Normal. 2 Theorem 15.4 1 1 2 The following properties hold: 1. If Z N (0; 1) and X = + 1=2 Z 2. If X N (; ), then = 1 2 then X N (; ). (X ) N (0; 1). N (; ) a is a vector of the same length as X , then aT X N (aT ; aT a). 3. If X 276 15. Multivariate Models 4. Let V = (X Then V )T 1 (X ): k . 2 Suppose we partition a random Normal vector X into two parts X = (Xa; Xb) We can similarly partition the the mean = (a ; b ) and the variance = aa ab : ba bb X N (; ). Then: (1) The marginal distribition of Xa is Xa N (a ; aa ). (2) The conditional distribition of Xb given Xa = xa is Theorem 15.5 Let Xb jXa = xa N ((xa ); (xa )) where (xa ) = b + ba aa1 (xa a ) (xa ) = bb ba aa1 ab : (15.9) (15.10) n from a N (; ), the log-likelihood is (up to a constant not depending on or ) Theorem 15.6 Given a random sample of size is given by n n n ( X )T (X ) tr( S ) 2 2 2 log det(): `(; ) = The mle 1 1 is b = n 1 S: b = X and n (15.11) 15.5 Appendix 15.5 277 Appendix Proof of Theorem 15.6. Denote the i random vector by X i. The log-likelihood is X n 1 X(X i )T (X i ): kn log(2 ) log det() `(; ) = f (X i; ; ) = 2 2 2 i i th 1 Now, X i (X i )T (X i ) = 1 = P X [(X i X ) + (X )]T [(X i X ) + (X )] i 1 X (X i X )T (X i X ) + n(X )T (X ) i 1 1 since i(X i X ) (X ) = 0. Also, notice that (X i )T (X i ) is a scalar, so 1 1 X i (X i )T (X i ) = X = X 1 i i tr (X i )T 1 (X i ) tr 1 (X i )(X i )T X )(X i )T " = tr 1 i = n tr S 1 # (X i and the conclusion follows. 15.6 Exercises 1. Prove Theorem 15.1. 2. Find the Fisher information matrix for the mle of a Multinomial. 3. Prove Theorem 15.5. 4. (Computer Experiment. Write a function to generate nsim observations from a Multinomial(n; p) distribution. 278 15. Multivariate Models 5. (Computer Experiment. Write a function to generate nsim observations from a Multivariate normal with given mean and covariance matrix . 6. (Computer Experiment. Generate 1000 random vectors from a N (; ) distribution where = 3 ; = 8 2 6 : 6 2 Plot the simulation as a scatterplot. Find the distribution of X jX = x using theorem 15.5. In particular, what is the formula for E (X jX = x )? Plot E (X jX = x ) on your scatterplot. Find the correlation between X and X . Compare this with the sample correlations from your simulation. Find a 95 per cent condence interval for . Estimate the covariance matrix . 2 1 1 2 1 1 2 1 1 1 2 7. Generate 100 random vectors from a multivariate Normal with mean (0; 2)T and variance 1 3 : 3 1 Find a 95 per cent condence interval for the correlation . What is the true value of ? This is page 279 Printer: Opaque this 16 Inference about Independence In this chapter we address the following questions: (1) How do we test if two random variables are independent? (2) How do we estimate the strength of dependence between two random variables? When Y and Z are not independent, we say that they are or or If Y and Z are associated, it does not imply that Y causes Z or that Z causes Y . If Y does cause Z then changing Y will change the distribution of Z , otherwise it will not change the distribution of Z . For example, quitting smoking Y will reduce your probability of heart disease Z . In this case Y does cause Z . As another example, owning a TV Y is associated with having a lower incidence of starvation Z . This is because if you own a TV you are less likely to live in an impoverished nation. But giving a starving person will not cause them to stop being hungry. In this case, Y and Z are associated but the relationship is not causal. dependent associated related. Recall that we write Y q Z to mean that Y and Z are independent. 280 16. Inference about Independence We defer detailed discussion of the important question of causation until a later chapter. 16.1 Two Binary Variables Suppose that Y and Z are both binary. Consider a data set (Y1 ; Z1 ); : : : ; (Yn ; Zn). Represent the data as a two-by-two table: Y =0 Y =1 Z = 0 X00 X01 X0 Z = 1 X10 X11 X1 X0 X1 n = X where the Xij represent counts: Xij = number of observations for which Y = i and Z = j: P The dotted subscripts denote sums. For example, Xi = j Xij . This is a convention we use throughout the remainder of the book. Denote the corresponding probabilities by: Y =0 Y =1 Z = 0 p00 p01 p0 Z = 1 p10 p11 p1 p0 p1 1 where pij = P(Z = i; Y = j ). Let X = (X00 ; X01 ; X10 ; X11 ) denote the vector of counts. Then X Multinomial(n; p) where p = (p00 ; p01 ; p10 ; p11 ). It is now convenient to introduce two new parameters. 16.1 Two Binary Variables Denition 16.1 The odds ratio is dened to be p00 p11 : p01 p10 (16.1) = log( ): (16.2) = The 281 log odds ratio is dened to be Theorem 16.2 The following statements are equivalent: 1. Y q Z . 2. = 1. 3. = 0. 4. For i; j 2 f0; 1g, pij = pi pj : (16.3) Now consider testing H0 : Y qZ versus H1 : Y q6 Z: First we consider the likelihood ratio test. Under H1 , X Multinomial(n; p) and the mle is pb = X=n. Under H0 , we again have that X Multinomial(n; p) but p is subject to the constraint pij = pi pj ; j = 0; 1: This leads to the following test. Theorem Let 16.3 (Likelihood Ratio Test for Independence in a 2-by-2 table) T =2 1 X 2 X i=0 X X Xij log ij : XiXj j =0 (16.4) Under H0 , T 21 . Thus, an approximate level test is obtained by rejecting H0 then T > 21; . 282 16. Inference about Independence Another popular test for independence is Pearson's 2 test. Theorem Let 16.4 (Pearson's test for Independence in a 2-by-2 table) 2 U= where 1 X 1 X (Xij i=0 j =0 Eij = Eij )2 Eij (16.5) Xi Xj : n Under H0 , U 21 . Thus, an approximate level test is obtained by rejecting H0 then U > 21; . Here is the intuition for the Pearson test. Under H0 , pij = pipj , so the maximum likelihood estimator of pij under H0 is pbij = pbi pbj = Xi Xj : n n Thus, the expected number of observations in the (i,j) cell is Eij = npbij = XiXj : n The statistic U compares the observed and expected counts. Example 16.5 The following data from Johnson and Johnson (1972) relate tonsillectomy and Hodgkins disease. (The data are actually from a case-control study; we postpone discussion of this point until the next section.) Hodgkins Disease No Disease Tonsillectomy 90 165 255 No Tonsillectomy 84 307 391 Total 174 472 646 We would like to know if tonsillectomy is related to Hodgkins disease. The likelihood ratio statistic is T = 14:75 and the pvalue is P(21 > 14:75) = :0001. The 2 statistic is U = 14:96 16.1 Two Binary Variables 283 and the p-value is P(21 > 14:96) = :0001. We reject the null hypothesis of independence and conclude that tonsillectomy is associated with Hodgkins disease. This does not mean that tonsillectomies cause Hodgkins disease. Suppose, for example, that doctors gave tonsillectomies to the most seriosuly ill patients. Then the assocation between tonsillectomies and Hodgkins disease may be due to the fact that those with tonsillectomies were the most ill patients and hence more likely to have a serious disease. We can also estimate the strength of dependence by estimating the odds ratio and the log-odds ratio . Theorem 16.6 The mle 's of b= and are X00 X11 ; b = log b: X01 X10 (16.6) The asymptotic stanrad errors (computed from the delta method) are r 1 1 1 1 b (b se ) = + + + (16.7) X00 X01 b ( b) = bse b (b se ): X10 X11 (16.8) Remark 16.7 For small sample sizes, b and b can have a very large variance. In this case, we often use the modied estimator X00 + 21 X11 + 12 b= : X01 + 21 X10 + 12 (16.9) Yet another test for indepdnence is the Wald test for = 0 b (b given by W = (b 0)=se ). A 1 condence interval for is z=2 seb (b ). A 1 condence interval for can be obtained b b ( b). Second, since in two ways. First, we could use b z=2 se = e we could use b (b exp b z=2 se ) : This second method is usually more accurate. (16.10) 284 16. Inference about Independence Example 16.8 In the previous example, b= and b 90 307 = 1:99 165 84 = log(1:99) = :69: So tonsillectomy patients were twice as likely to have Hodgkins disease. The standard error of b is r 1 1 1 1 + + + = :18: 90 84 165 307 The Wald statistic is W = :69=:18 = 3:84 whose p-value is P(jZ j > 3:84) = :0001, the same as the other tests. A 95 per cent condence interval for is b 2(:18) = (:33; 1:05). A 95 per cent condence interval for is (e:33 ; e1:05 ) = (1:39; 2:86). 16.2 Interpreting The Odds Ratios Suppose event A as probability P (A). The odds of A are dened as P (A) : odds(A) = 1 P (A) It follows that odds(A) P(A) = : 1 + odds(A) Let E be the event that someone is exposed to something (smoking, radiation, etc) and let D be the event that they get a disease. The odds of getting the disease given that you are exposed are P(D jE ) odds(DjE ) = 1 P(DjE ) and the odds of getting the disease given that you are not exposed are P(D jE c ) : odds(DjE c) = 1 P(DjE c) 16.2 Interpreting The Odds Ratios 285 The odds ratio is dened to be = odds(DjE ) : odds(DjE c) If = 1 then disease probability is the same for exposed and unexposed. This implies that these events are independent. Recall that the log-odds ratio is dened as = log( ). Independence corresponds to = 0. Consider this table of probabilities: Dc E c p00 E p10 p0 D p01 p0 p11 p1 p1 1 Denote the data by Dc D E c X00 X01 X0 E X10 X11 X1 X0 X1 X Now P (D jE ) = p p+11p 10 11 and P(DjE c) = p01 p00 + p01 and so odds(DjE ) = and therefore, p11 p and odds(DjE c) = 01 p10 p00 = p11 p00 : p01 p10 To estimate the parameters, we have to rst consider how the data were collected. There are three methods. Multinomial Sampling. We draw a sample from the population and, for each person, record their exposure and disease status. In this case, X = (X00 ; X01 ; X10 ; X11 ) Multinomial(n; p). 286 16. Inference about Independence We then estimate the probabilities in the table by pbij = Xij =n and b = pb11 pb00 = X11 X00 : pb01 pb10 X01 X10 We get some exposed and unexposed people and count the number with disease in each group. Thus, Prospective Sampling. (Cohort Sampling). X01 X11 Binomial(X0 ; P (DjE c)) Binomial(X1 ; P (DjE )): We should really write x0 and x1 instead of X0 and X1 since in this case, these are xed not random but for notational simplicity I'll keep using capital letters. We can estimate P (DjE ) and P (DjE c) but we cannot estimate all the probabilities in the table. Still, we can estimate since is a function of P (DjE ) and P (DjE c). Now X Pb(DjE ) = 11 X1 and X Pb(DjE c) = 01 : X0 Thus, b = X11 X00 X01 X10 just as before. Case-Control (Retrospective) Sampling. Here we get some diseased and non-diseased people and we observe how many are exposed. This is much more eÆcient if the disease is rare. Hence, X10 X11 Binomial(X0 ; P (E jDc)) Binomial(X1 ; P (E jD)): From these data we can estimate P(E jD) and P(E jDc). Surprisingly, we can also still estimate . To understand why, note 16.2 Interpreting The Odds Ratios that P(E jD) = p p+11p ; 01 11 1 P(E jD) = p01 p01 + p11 By a similar argument, odds(E jDc) = 287 ; odds(E jD) = p11 : p01 p10 : p00 Hence, odds(E jD) p11 p00 = = : odds(E jDc) p01 p10 From the data, we form the following estimates: Pb(E jD) = Therefore, X11 X [ X [ X ; 1 Pb(E jD) = 01 ; odds( E jD) = 11 ; odds( E jDc) = 10 : X1 X1 X01 X00 b= X00 X11 : X01 X10 So in all three data collection methods, the estimate of out to be the same. turns It is tempting to try to estimate P(DjE ) P(DjE c). In a case-control design, this quantity is not estimable. To see this, we apply Bayes' theorem to get P(E jD )P(D ) P(E c jD )P(D ) P(D jE ) P(D jE c ) = : P(E ) P(E c ) Because of the way we obtained the data, P(D) is not estimable from the data. However, we can estimate = P(DjE )=P(DjE c), which is called the , under the relative risk rare disease assumption. Theorem 16.9 Let = P(DjE )=P(DjE c). Then as P(D) ! 0. !1 288 16. Inference about Independence Thus, under the rare disease assumption, the relative risk is approximately the same as the odds ratio and, as we have seen, we can estimate the ods ratio. In a randomized experiment, we can interpret a strong association, that is 6= 1, as a causal relationship. In an observational (non-randomized) study, the association can be due to other unobserved variables. We'll discuss causation in more detail later. confounding 16.3 Two Discrete Variables Now suppose that Y 2 f1; : : : ; I g and Z 2 f1; : : : ; J g are two discrete variables. The data can be represented as an I by J table of counts: Z=1 .. . Z=i .. . Z=I Y =1 Y =2 X11 X12 .. . .. . Y = j Y = J X1j X1J X1 .. . .. . .. . .. . .. . .. . .. . Xi1 Xi2 Xij XiJ Xi XI 1 X1 XI 2 X2 XIj Xj XIJ XJ XI .. . .. . Consider testing H0 : Y .. . q Z versus H1 : Y q6 Z . .. . .. . n 16.3 Two Discrete Variables Theorem 16.10 Let T =2 I X J X i=1 289 X X Xij log ij : XiXj j =1 (16.11) The limiting distribution of T under the null hypothesis of independence is 2 where = (I 1)(J 1). Pearson's 2 test statistic is U= I X J X (nij i=1 j =1 Eij )2 Eij : (16.12) Asymptotically, under H0 , U has a 2 distribution where = (I 1)(J 1). Example 16.11 These data are from a study by Hancock et al (1979). Patients with Hodkins disease are classied by their response to treatment and by histological type. Type LP NS MC LD Positive Response Partial Response No Response 74 18 12 104 68 16 12 96 154 54 58 266 18 10 44 72 The 2 test statistic is 75.89 with 2 3 = 6 degrees of freedom. The p-value is P(26 > 75:89) 0. The likelihood ratio test statistic is 68.30 with 2 3 = 6 degrees of freedom. The p-value is P(26 > 68:30) 0. Thus there is strong evidence that reponse to treatment and histological type are associated. There are a variety of ways to quantify the strength of dependence between two discrete variables Y and Z . Most of them are not very intuitive. The one we shall use is not standard but is more interpretable. 290 16. Inference about Independence We dene Æ (Y; Z ) = max jPY;Z (Y A;B 2 A; Z 2 B ) P Y (Y 2 A) PZ (Z 2 B )j (16.13) where the maximum is over all pairs of events A and B . Theorem 16.12 Properties of Æ: 1. 0 Æ (Y; Z ) 1. 2. Æ (Y; Z ) = 0 if and only if Y q Z. 3. The following identity holds: I X J 1X Æ (X; Y ) = p 2 i=1 j =1 ij 4. The mle (16.14) of Æ is Æ (X; Y ) = where pi pj : I J 1 X X pb 2 i=1 j =1 ij pbi pbj (16.15) Xij X X ; pbi = i ; pbj = j : n n n The interpretation of Æ is this: if one person makes probability pbij = statements assuming independence and another person makes probability statements without assuming independence, their probability statements may dier by as much as Æ . Here is a suggested scale for interpreting Æ : 0 < Æ :01 negligible association :01 < Æ :05 non-negligible association :05 < Æ :10 substantial association Æ > :10 very strong association A condence interval for Æ can be obtained by bootstrapping. The steps for bootstrapping are: 16.4 Two Continuous Variables 291 1. Draw X Multinomial(n; pb); 2. Compute pbij ; pbi ; pbj . 3. Compute Æ . 4. Repeat. Now we use any of the methods we learned earlier for constructing bootstrap condence intervals. However, we should not use a Wald interval in this case. The reason is that if Y q Z then Æ = 0 and we are on the boundary of the parameter space. In this case, the Wald method is not valid. Example 16.13 Returning to Example 16.11 we nd that Æb = :11. Using a pivotal bootstrap with 10; 000 bootstrap samples, a 95 per cent condence interval for Æ is (.09,.22). Our conclusion is that the association between histological type and response is substantial. 16.4 Two Continuous Variables Now suppose that Y and Z are both continuous. If we assume that the joint distribution of Y and Z is bivariate Normal, then we mesure the dependence between Y and Z by means of the correaltion coeÆcient . Tests, estimates and condence intervals for in the Normal case are given in the previous chapter. If we do not assume Normality then we need a nonparametric method for assessing dependence. Recall that the correaltion is E ((X1 1 )(X2 2 )) = : 1 2 A nonparametric estimator of is the plug-in estimator which is b = Pn (X1i X 1 )(X2i X 2 ) qP i=1 Pn n 2 2 i=1 (X1i X 1 ) i=1 (X2i X 2 ) 292 16. Inference about Independence which is just the sample correlation. A condence interval can be constructed using the bootstrap. A test for = 0 can be based on the Wald test using the bootstrap to estimate the standard error. The plug-in approach is useful for large samples. For small samples, we measure the correlation using the bS . We simply replace the data by their ranks { ranking each variable separately { then we compute the correaltion coeÆcient of the ranks. To test the null hypothesis that S = 0 we need the distribution of bS under the null hypothesis. This can be obtained easily by simulation. We x the ranks of the rst variable as 1; 2; : : : ; n. The ranks of the second variable are chosen at random from the set of n! possible orderings. Then we compute the correlation. This is repeated many times and the resulting distribution P0 is the null distribution of bS . The p-value for the test is P0 (jRj > jbS j) where R is drawn from the null distribution P0 . correaltion coeÆcient Spearman rank Example 16.14 The following data (Snedecor and Cochran, 1980, p. 191) are systolic blood pressure X1 and are diastolic blood pressure X2 in millimeters: X1 100 105 110 110 120 120 125 130 130 150 170 195 X2 65 65 75 70 78 80 75 82 80 90 95 90 The sample correlation is b = :88. The bootstrap standard error is .046 and the Wald test statistic is :88=:046 = 19:23. The p-value is near 0 giving strong evidence that the correlation is not 0. The 95 per cent pivotal bootstrap condence interval is (.78,.94). Because the sample size is small, consider Spearman's rank correlation. In ranking the data, we will use average ranks if there are ties. So if the third and fourth lowest numbers are the same, they each get rank 3.5. The ranks of the data are: 16.5 One Continuous Variable and One Discrete 293 X1 1 2 3.5 3.5 5.5 5.5 7 8.5 8.5 10 11 12 X2 1.5 1.5 4.5 3 6 7.5 4.5 9 7.5 10.5 12 10.5 The rank correlation is bS = :94 and the p-value for testing the null hypothesis that there is no correaltion is P0 (jRj > :94) 0 which is was obtained by simulation. 16.5 One Continuous Variable and One Discrete Suppose that Y 2 f1; : : : ; I g is discrete and Z is continuous. Let Fi (z ) = P(Z z jY = i) denote the cdf of Z conditional on Y = i. Theorem 16.15 When Y 2 f1; : : : ; I g is discrete and Z is con- tinuous, then Y q Z if and only if F1 = = FI . It follows from the previous theorem that to test for independence, we need to test H0 : F1 = = FI versus H1 : not H0 : For simplicity, we consider the case where I = 2. To test the null hypothesis that F1 = F2 we will use the . Let n1 denote the nimber of observations for which Yi = 1 and let n2 denote the nimber of observations for which Yi = 2. Let two sample Kolmogorov-Smirnov test Fb1 (z ) = and Fb2 (z ) = n 1 X n1 i=1 n 1 X n2 i=1 I (Zi z )I (Yi = 1) I (Zi z )I (Yi = 2) denote the empirical distribution function of Z given Y = 1 and Y = 2 respectively. Dene the test statistic D = sup jFb1 (x) Fb2 (x)j: x 294 16. Theorem Inference about Independence 16.16 Let H (t) = 1 2 1 X j =1 ( 1)j 1 e Under the null hypthesis that F1 = F2 , r lim P n!1 2j 2 t2 : n1 n2 D t = H (t): n1 + n2 It follows from the theorem that an approximate level test is obtained by rejecting H0 when r 16.6 n1 n2 D > H 1 (1 ): n1 + n 2 Bibliographic Remarks Johnson, S.K. and Johnson, R.E. (1972). New England Journal of Medicine. 287. 1122-1125. Hancock, B.W. (1979). Clinical Oncology, 5, 283-297. 16.7 Exercises 1. Prove Theorem 16.2. 2. Prove Theorem 16.3. 3. Prove Theorem 16.9. 4. Prove equation (16.14). The data here an approximate creation using information in article. are rethe the 5. The New York Times (January 8, 2003, page A12) reported the following data on death sentencing and race, from a study in Maryland: Death Sentence No Death Sentence Black Victim 14 641 White Victim 62 594 16.7 Exercises 295 Analyze the data using the tools from this Chapter. Interpret the results. Explain why, based only on this information, you can't make causal conclusions. (The authors of the study did use much more information in their full report.) 6. Analyze the data on the variables Age and Financial Status from: http://lib.stat.cmu.edu/DASL/Datales/montanadat.html 7. Estimate the correlation between temperature and latitude using the data from http://lib.stat.cmu.edu/DASL/Datales/USTemperatures.html Use the correlation coeÆcient and the Spearman rank correaltion. Provide estimates, tests and condence intervals. 8. Test whether calcium intake and drop in blood pressure are associated. Use the data in http://lib.stat.cmu.edu/DASL/Datales/Calcium.html This is page 297 Printer: Opaque this 17 Undirected Graphs and Conditional Independence Graphical models are a class of multivariate statistical models that are useful for representing independence relations. They are also useful develop parsimonious models for multivariate data. To see why parsimonious models are useful in the multivariate setting, consider the problem of estimating the joint distribution of several discrete random variables. Two binary variables Y1 and Y2 can be represented as a two-by-two table which corresponds to a multinomial with four categories. Similarly, k binary variables Y1 ; : : : ; Y correspond to a multinomial with N = 2 categories. When k is even moderately large, N = 2 will be huge. It can be shown in that case that the mle is a poor estimator. The reason is that the data are sparse: there are not enough data to estimate so many parameters. Graphical models often require fewer parameters and may lead to estimators with smaller risk. There are two main types of graphical models: undirected and directed. Here, we introduce undirected graphs. We'll discuss directed graphs later. k k k 298 17. Undirected Graphs and Conditional Independence 17.1 Conditional Independence Underlying graphical models is the concept of conditional independence. Denition 17.1 Let X , Y and Z be discrete random variables. We say that X and Y are conditionally independent given Z , written X q Y j Z , if P(X = x; Y = j y Z = z) = P(X = xjZ = z )P(Y = j y Z = z) (17.1) for all x; y; z . If X , Y and Z are continuous random variables, we say that X and Y are conditionally independent given Z if fX;Y for all x, y j (x; y jz ) = f j (xjz )f j (y jz ): Z X Z Y Z and z . Intuitively, this means that, once you know Z , Y provides no extra information about X . The conditional independence relation satises some basic properties. Theorem 17.2 The following implications hold: =) X q Y j Z and U = h(X ) =) X q Y j Z and U = h(X ) =) X q Y j Z and X q W j(Y; Z ) =) X q Y j Z and X q Z j Y =) X q j Y Z Y U X X X q j q j q j( ) q( )j q( ) X Z Y Z Y Z; U W; Y Z Y; Z : The last property requires the assumption that all events have positive probability; the rst four do not. 17.2 Undirected Graphs 299 Y b b X b FIGURE 17.1. A graph with vertices E = f(X; Y ); (Y; Z )g. 17.2 V = Z fX; Y ; Z g. The edge set is Undirected Graphs An undirected graph G = (V; E ) has a nite set V of vertices (or nodes) and a set E of edges (or arcs) consisting of pairs of vertices. The vertices correspond to random variables X; Y; Z; : : : and edges are written as unordered pairs. For example, (X; Y ) 2 E means that X and Y are joined by an edge. An example of a graph is in Figure 17.1. Two vertices are adjacent, written X Y , if there is an edge between them. In Figure 17.1, X and Y are adjacent but X and Z are not adjacent. A sequence X0 ; : : : ; X is called a path if X 1 X for each i. In Figure 17.1, X; Y; Z is a path. A graph is complete if there is an edge between every pair of vertices. A subset U V of vertices together with their edges is called a n i i subgraph. If are three distinct subsets if V , we say that C separates A and B if every path from a variable in A to a variable in B intersects a variable in C . In Figure 17.2 fY; W g and fZ g are separated by fX g. Also, W and Z are separated by fX; Y g. A; B and C 300 17. Undirected Graphs and Conditional Independence FIGURE 17.2. Z W b b X Y b b Z fY; W g and fZ g fX; Y g. are separated by fX g. Also, W and are separated by Y b X b b FIGURE 17.3. 17.3 X Z q Z jY . Probability and Graphs Let V be a set of random variables with distribution P. Construct a graph with one vertex for each random variable in V . Suppose we omit the edge between a pair of variables if they are independent given the rest of the variables: no edge between X and Y () X q jrest Y where \rest" refers to all the other variables besides X and Y . This type of graph is called a pairwise Markov graph. Some examples are shown in Figures 17.3, 17.4 17.6 and 17.5. The graph encodes a set of pairwise conditional independence relations. These relations imply other conditional independence 17.3 Probability and Graphs 301 Y b X b b Z FIGURE 17.4. No implied independence relations. X b bW Y b b Z and Y FIGURE 17.5. b X X q Z jfY; W g b Y q W jfX; Z g. b b Z W FIGURE 17.6. Pairwise independence implies that But is X q Z jY ? X q Z jfY ; W g. 302 17. Undirected Graphs and Conditional Independence relations. How can we gure out what they are? Fortunately, we can read these other conditional independence relations directly from the graph as well, as is explained in the next theorem. Theorem 17.3 Let G=( ) be a pairwise Markov graph for a distribution P. Let A; B and C be distinct subsets of V such that C separates A and B . Then A q B jC . V; E Remark 17.4 If A and B are not connected (i.e. there is no path from A to B ) then we may regard A and B as being separated by the empty set. Then Theorem 17.3 implies that A q B . The independence condition in Theorem 17.3 is called the global Markov property. We thus see that the pairwise and global Markov properties are equivalent. Let us state this more precisely. Given a graph G , let Mpair (G ) be the set of distributions which satisfy the pairwise Markov property: thus P 2 Mpair (G ) if, under P , X q Y jrest if and only if there is no edge between X and Y . Let Mglobal (G ) be the set of distributions which satisfy the global Markov property: thus P 2 Mpair (G ) if, under P, A q B jC if and only if C separates A and B . Theorem 17.5 Let G be a graph. Then, Mpair (G ) = Mglobal (G ). This theorem allows us to construct graphs using the simpler pairwise property and then we can deduce other independence relations using the global Markov property. Think how hard this would be to do algebraically. Returning to 17.6, we now see that X q Z jY and Y q W jZ . Example 17.6 Figure 17.7 implies that (Y; Z ). Example 17.7 Figure 17.8 implies that j Z Y . X X q Y , X q j( W q and X ) and X Z Y; Z q q 17.3 Probability and Graphs Y b X b b FIGURE 17.7. X X b qY, X qZ and X Z q (Y; Z ). Y b bW b Z FIGURE 17.8. X q W j(Y; Z ) and X q Z jY . 303 304 17.4 17. Undirected Graphs and Conditional Independence Fitting Graphs to Data Given a data set, how do we nd a graphical model that ts the data. Some authors have devoted whole books to this subject. We will only treat the discrete case and we will consider a method based on log-linear models which are the subject of the next chapter. 17.5 Bibliographic Remarks Thorough treatments of undirected graphs can be found in Whittaker (1990) and Lauritzen (1996). Some of the exercises below are adapted from Whittaker (1990). 17.6 Exercises 1. Consider random variables (X1 ; X2 ; X3 ). In each of the following cases, draw a graph that has the given independence relations. (a) X1 q X3 j X2 . (b) X1 q X2 j X3 and X1 q X3 j X2 (c) X1 q X2 j X3 and X1 q X3 j X2 . and X2 q X3 j X1 . 2. Consider random variables (X1 ; X2 ; X3 ; X4 ). In each of the following cases, draw a graph that has the given independence relations. (a) X1 q X3 j X2 ; X4 and X1 q X4 j X2 ; X3 and X2 q X4 j X1 ; X3 . 17.6 Exercises X1 X2 b b 305 X4 b b X3 FIGURE 17.9. X1 X2 b b X3 b X4 b FIGURE 17.10. (b) X1 q X2 j X3 ; X4 and X1 q X3 j X2 ; X4 and X2 q X3 j X1 ; X4 . (c) X1 q X3 j X2 ; X4 and X2 q X4 j X1 ; X3 . 3. A conditional independence between a pair of variables is minimal if it is not possible to use the Separation Theorem to eliminate any variable from the conditioning set, i.e. from the right hand side of the bar (Whittaker 1990). Write down the minimal conditional independencies from: (a) Figure 17.9; (b) Figure 17.10; (c) Figure 17.11; (d) Figure 17.12. 4. Here are breast cancer data on diagnostic center (X1 ), nuclear grade (X2 ), and survival (X3 ): 306 17. Undirected Graphs and Conditional Independence X3 X4 b bX b bX 2 1 FIGURE 17.11. X1 b X2 b b X3 b b b X4 X5 X6 FIGURE 17.12. 17.6 Exercises 307 malignant malignant benign benign X3 died survived died survived Boston 35 59 47 112 Glamorgan 42 77 26 76 X2 X1 (a) Treat this as a multinomial and nd the maximum likelihood estimator. (b) If someone has a tumour classied as benign at the Glamorgan clinic, what is the estimated probability that they will die? Find the standard error for this estimate. (c) Test the following hypotheses: q 2j 1 q 3j 2 q 3j X1 X X3 X X X2 X X X1 versus versus versus q= 2j = 3j 1 q = 3j 2 q X1 X X3 X X X2 X X X1 Based on the results of your tests, draw and interpret the resulting graph. 18 Loglinear Models In this chapter we study loglinear models which are useful for modelling multivariate discrete data. There is a strong connection between loglinear models and undirected graphs. Parts of this Chapter draw on the material in Whittaker (1990). 18.1 The Loglinear Model Let X = (X1 ; : : : ; Xm) be a random vector with probability function f (x) = P(X = x) = P(X1 = x1; : : : ; Xm = xm ) where x = (x1; : : : ; xm ). Let rj be the number of values that Xj takes. Without loss of generality, we can assume that Xj 2 f0; 1; : : : ; rj 1g. Suppose now that we have n such random vectors. We can think of the data as a sample from a Multinomial with N = r1 r2 rm categories. The data can be represented as counts in a r1 r2 rm table. Let p = (p1; : : : ; pN ) denote the multinomial parameter. This is page 309 Printer: Opaque this 310 18. Loglinear Models Let S = f1; : : : ; mg. Given a vector x = (x1 ; : : : ; xm ) and a subset A S , let xA = (xj : j 2 A). For example, if A = f1; 3g then xA = (x1; x3 ). Theorem 18.1 The joint probability function f (x) of a single random vector X = (X1 ; : : : ; Xm ) can be written as X log f (x) = (18.1) A (x) AS where the sum is over all subsets A of S 's satisfy the following conditions: = f1; : : : ; mg and the 1. ; (x) is a constant; 2. For every A S , the rest of the x0j s. A (x) is only a function of xA and not 3. If i 2 A and xi = 0, then (x) = 0. The formula in equation (18.1) is called the log-linear expansion of f . Note that this is the probability function for a single draw. Each A (x) will depend on some unknown parameters A. Let = (A : A S ) be the set of all these parameters. We will write f (x) = f (x; ) when we want to estimate the dependence on the unknown parameters . In terms of the multinomial, the parameter space is P= n A p = (p1 ; : : : ; pN ) : pj 0; N X j =1 pj o =1 : This is an N 1 dimensional space. In the log-linear represenation, the parameter space is n o = = (1 ; : : : ; N ) : = (p); p 2 P where (p) is the set of values associated with p. The set is a N 1 dimensinal surface in R N . We can always go back and forth betwee the two parameterizations we can write = (p) and p = p( ). 18.1 The Loglinear Model Example 18.2 Let X 311 Bernoulli(p) where 0 < p < 1. We can write the probability mass function for X as f (x) = px (1 p)1 x = px1 p12 for x = 0; 1, where p1 = p and p2 = 1 x p. Hence, log f (x) = ; (x) + 1 (x) where ; (x) = log(p2) p1 1 (x) = x log p : 2 Notice that ; (x) is a constant (as a function of x) and 1 (x) = 0 when x = 0. Thus the three conditions of the Theorem hold. The loglinear parameters are 0 = log(p2 ); 1 = log p1 : p2 The original, multinomial parameter space is P = f(p1 ; p2 ) pj 0; p1 + p2 = 1g. The log-linear parameter space is = f(0 ; 1) 2 R 2 : e0 +1 + e0 : = 1:g Given (p1 ; p2 ) we can solve for (0 ; 1 ). Conversely, given (0 ; 1 ) we can solve for (p1 ; p2 ). Example 18.3 Let X = (X1; X2) where X1 2 f0; 1g and X2 2 f0; 1; 2g. The joint distribution of n such random vectors is a multinomial with 6 categories. The multinomial parameters can be written as a 2-by-3 table as follows: multinomial x2 0 1 2 x1 0 p00 p01 p02 1 p10 p11 p12 312 18. Loglinear Models The n data vectors can be summarized as counts: data x2 0 1 2 x1 0 C00 C01 C02 1 C10 C11 C12 For x = (x1 ; x2 ), the log-linear expansion takes the form log f (x) = ; (x) + 1 (x) + 2 (x) + 12 (x) where ; (x) = log p00 p10 1 (x) = x1 log p 2 (x) 12 (x) 00 = I (x2 = 1) log p01 p00 = I (x1 = 1; x2 = 1) log + I (x2 = 2) log p11 p00 p01 p10 p02 p00 + I (x1 = 1; x2 = 2) log Convince yourself that the three conditions on the 's of the theorem are satised. The six parameters of this model are: 1 = log p00 4 = log p02 p00 2 = log 5 = log p10 p00 p11 p00 p01 p10 3 = log 6 = log p01 p00 p12 p00 p02 p10 : The next Theorem gives an easy way to check for conditional independence in a loglinear model. Theorem 18.4 Let (Xa ; Xb ; Xc) be a partition of a vectors (X1 ; : : : ; Xm ). Then Xb q Xc jXa if and only if all the -terms in the log-linear expansion that have at least one coordinate in b and one coordinate in c are 0. To prove this Theorem, we will use the following Lemma whose proof follows easily from the denition of conditional independence. p12 p00 : p02 p10 18.2 Graphical Log-Linear Models 313 Lemma 18.5 A partition (Xa ; Xb ; Xc) satises Xb q XcjXa if and only if f (xa ; xb ; xc ) = g (xa ; xb )h(xa ; xc ) for some functions g and h (Theorem 18.4.) Suppose that t is S 0 whenever tShas coordinates in b and c. Hence, t is 0 if t 6 a b or t 6 a c. Therefore X X X log f (x) = S t (x) + S t (x) t (x): Proof. ta b ta ta c Exponentiating, we see that the joint density is of the form g (xa ; xb )h(xa ; xc ). By Lemma 18.5, Xb q Xc jXa . The converse follows by reversing the argument. 18.2 Graphical Log-Linear Models A log-linear model is graphical if missing terms correspond only to conditional independence constraints. Denition 18.6 Let log f (x) = P AS A (x) be a log-linear model. Then f is graphical if all -terms are non-zero except for any pair of coordinates not in the edge set for some graph G . In other words, A (x) = 0 if and only if fi; j g A and (i; j ) is not an edge. Here is way to think about the denition above: If you can add a term to the model and the graph does not change, then the model is not graphical. Example 18.7 Consider the graph in Figure 18.1. is The graphical log-linear model that corresponds to this graph log f (x) = + ; + 1 (x) + 2 (x) + 3 (x) + 4 (x) + 5 (x) 12 (x) + 23 (x) + 25 (x) + 34 (x) + 35 (x) + 45 (x) + 235 (x) + 345 (x): 314 18. Loglinear Models X5 X4 b b b X1 X2 X3 b b FIGURE 18.1. Graph for Example 18.7. Let's see why this model is graphical. The edge (1; 5) is missing in the graph. Hence any term containing that pair of indices is omitted from the model. For example, 15 ; 125 ; 135 ; 145 ; 1235 ; 1245 ; 1345 ; 12345 are all omitted. Similarly, the edge (2; 4) is missing and hence 24 ; 124 ; 234 ; 245 ; 1234 ; 1245 ; 2345 ; 12345 are all omitted. There are other missing edges as well. You can check that the model omits all the corresponding terms. Now consider the model log f (x) = + ; (x) + 1 (x) + 2 (x) + 3 (x) + 4 (x) + 5 (x) 12 (x) + 23 (x) + 25 (x) + 34 (x) + 35 (x) + 45 (x): This is the same model except that the three way interactions were removed. If we draw a graph for this model, we will get the same graph. For example, no terms contain (1; 5) so we omit the edge between X1 and X5 . But this is not graphical since it has extra terms omitted. The independencies and graphs for the two models are the same but the latter model has other constraints besides conditional independence constraints. This is not a bad 18.3 Hierarchical Log-Linear Models 315 thing. It just means that if we are only concerned about presence or absence of conditional independences, then we need not consider such a model. The presence of the three-way interaction 235 means that the strength of association between X2 and X3 varies as a function of X5 . Its absence indicates that this is not so. 18.3 Hierarchical Log-Linear Models There is a set of log-linear models that is larger than the set of graphical models and that are used quite a bit. These are the hierarchical log-linear models. Denition 18.8 A log-linear model is hierarchical if 0 and a t implies that t = 0. a = Lemma 18.9 A graphical model is hierarchical but the reverse need not be true. Example 18.10 Let log f (x) = ; (x) + 1(x) + 2 (x) + 3 (x) + 12 (x) + 13 (x): The model is hierarchical; its graph is given in Figure 18.2. The model is graphical because all terms involving (1,3) are omitted. It is also hierarchical. Example 18.11 Let log f (x) = ; (x)+ 1 (x)+ 2 (x)+ 3 (x)+ 12 (x)+ 13 (x)+ 23 (x): The model is hierarchical. It is not graphical. The graph corresponding to this model is complete; see Figure 18.3. It is not graphical because 123 (x) = 0 which does not correspond to any pairwise conditional independence. 316 18. Loglinear Models b b b X1 X2 X3 FIGURE 18.2. Graph for Example 18.10. X1 X2 b b b X3 FIGURE 18.3. The graph is complete. The model is hierarchical but not graphical. 18.4 Model Generators b b b X1 X2 X3 317 FIGURE 18.4. The model for this graph is not hierarchical. Example 18.12 Let log f (x) = ; (x) + 3 (x) + 12 (x): The graph corresponding is in Figure 18.4. This model is not hierarchical since 2 = 0 but 12 is not. Since it is not hierarchical, it is not graphical either. 18.4 Model Generators Hierarchical models can be written succinctly using generators. This is most easily explained by example. Suppose that X = (X1 ; X2 ; X3 ). Then, M = 1:2 + 1:3 stands for log f = ; + 1 + 2 + 3 + 12 + 13 : The formula M = 1:2+1:3 says: \include 12 and 13 ." We have to also include the lower order terms or it won't be hierarchical. The generator M = 1:2:3 is the saturated model log f = ; + 1 + 2 + 3 + 12 + 13 + 23 + 123 : The saturated models corresponds to tting an unconstrained multinomial. Consider M = 1 + 2 + 3 which means log f = ; + 1 + 2 + 3: 318 18. Loglinear Models 1.2 1+2 1 2 ; FIGURE 18.5. The lattice with two variables. This is the mutual independence model. Finally, consider M = 1:2 which has log-linear expansion log f = ; + 1 + 2 + 12 : This model makes X3 jX2 = x2 ; X1 = x1 a uniform distribution. 18.5 Lattices Hierarchical models can be organized into something called a lattice. This is the set of all hierarchical models partially ordered by inclusion. The set of all hierarchical models for two variables can be illustrated as in Figure 18.5. M = 1:2 is the saturated model, M = 1 + 2 is the independence model, M = 1 is independence plus X2jX1 is uniform, M = 2 is independence plus X1 jX2 is uniform, M = 0 is the uniform distribution. The lattice of trivariate models is shown in gure 18.6. 18.6 Fitting Log-Linear Models to Data Let denote all the parameters in a log-linear model M . The loglikelihood for is `( ) = X j xj log pj ( ) 18.6 Fitting Log-Linear Models to Data saturated (graphical) 1.2.3 two-way 1.2 + 1.3 + 2.3 319 graphical 1.2 + 1.3 1.2 + 2.3 1.3 + 2.3 graphical 1.2 + 3 1.3 + 2 2.3 + 1 mutual independence (graphical) conditional uniform independence uniform 1+2+3 1.2 1.3 2.3 1+2 1+3 2+3 1 2 3 ; FIGURE 18.6. The lattice of models for three variables. 320 18. Loglinear Models where the sum is over the cells and p( ) denotes the cell probabilities corresponding to . The mle b generally has to be found numerically. The model with all possible -terms is called the saturated model. We can also t any sub-model which corresponds to setting some subset of terms to 0. Denition 18.13 For any submodel M , dene the deviance dev(M ) by dev(M ) = 2(b̀sat b̀M ) where b̀sat is the log-likelihood of the saturated model evaluated at the mle and b̀M is the log-likelihood of the model M evaluated at its mle . Theorem for 18.14 The deviance is the likelihood ratio test statistic H0 : the model is M versus H1 : the model is not M: d Under H0 , dev (M ) ! 2 with degrees of freedom equal to the dierence in the number of parameters between the saturated model and M . One way to nd a good model is to use the deviance to test every sub-model. Every model that is not rejected by this test is then considered a plausible model. However, this is not a good strategy for two reasons. First, we will end up doing many tests which means that there is ample opportunity for making Type I and Type II errors. Second, we will end up using models where we failed to reject H0. But we might fail to reject H0 due to low power. The result is that we end up with a bad model just due to low power. There are many model searching strategies. A common approach is to use some form of penalized likelihood. One version of penalized is the AIC that we used in regression. For any model 18.6 Fitting Log-Linear Models to Data M dene 321 AIC(M ) = 2 jM j (18.2) where jM j is the number of parameters. Here is the explanation of where AIC comes from. Consider a set of models fM1; M2 ; : : :g. Let fbj (x) denote the estimated probability function obtained by using the maximum likelihood estimator ofr mode Mj . Thus, fbj (x) = fb(x; bj ) where bj is the mle of the set of parameters j for model Mj . We will use the loss function D(f; fb) where X f (x) f (x) log D(f; g ) = g (x) x b̀(M ) is the Kullback-Leibler distance between two probability functions. The corresponding risk function is R(f;P fb) = E (D(f; fb). Notice that D(f; fb) = c A(f; fb) where c = x f (x) log f (x) deos not depend on fb and A(f; fb) = X x f (x) log fb(x): Thus, minimizing the risk is equivalent to maximizing a(f; fb) E (A(f; fb)). It is tempting to estimate a(f; fb) by Px fb(x) log fb(x) but, just as the training error in regression is a highly P bbiased estimate of prediction risk, it is also the case that x f (x) log fb(x) is a highly biased estimate of a(f; fb). In fact, the bias is approximately equal to jMj j. Thus: Theorem 18.15 AIC (Mj ) is an approximately unbiased estimate of a(f; fb). After nding a \best model" this way we can draw the corresponding graph. We can also check the overall t of the selected model using the deviance as described above. 322 18. Loglinear Models Example 18.16 Data on breast cancer from Morrison et al (1973) were presented in homework question 4 of Chapter 17. The data are on diagnostic center (X1 ), nuclear grade (X2 ), and survival (X3 ): (Morrison et al 1973): X2 malignant malignant benign benign X3 died survived died survived X1 Boston 35 59 47 112 Glamorgan 42 77 26 76 The saturated log-linear model is: CoeÆcient Standard Error Wald Statistic p-value (Intercept) 3.56 0.17 21.03 center 0.18 0.22 0.79 grade 0.29 0.22 1.32 survival 0.52 0.21 2.44 centergrade -0.77 0.33 -2.31 centersurvival 0.08 0.28 0.29 gradesurvival 0.34 0.27 1.25 centergradesurvival 0.12 0.40 0.29 The best sub-model, selected using AIC and backward searching is 0.00 *** 0.42 0.18 0.01 * 0.02 * 0.76 0.20 0.76 CoeÆcient Standard Error Wald Statistic p-value (Intercept) 3.52 0.13 25.62 < 0.00 *** center 0.23 0.13 1.70 0.08 grade 0.26 0.18 1.43 0.15 survival 0.56 0.14 3.98 6.65e-05 *** centergrade -0.67 0.18 -3.62 0.00 *** gradesurvival 0.37 0.19 1.90 0.05 The graph for this model M is shown in Figure 18.7. To test the t of this model, we compute the deviance of M which is 0.6. The appropriate 2 has 8 6 = 2 degrees of freedom. The p-value is P(22 > :6) = :74. So we have no evidence to suggest that the model is a poor t. 18.6 Fitting Log-Linear Models to Data Center Grade 323 Survival FIGURE 18.7. The graph for Example 18.16. The example above is a toy example. With so few variables, there is little to be gained by searching for a best model. The overall multinomial mle would be suÆcient. More importantly, we are probably interested in predicting survivial from grade and center in which case this should really be treated as a regresion problem with outcome Y = survival and covariates center and grade. Example 18.17 Here is a synthetic example. We generate n = 100 random vectors X = (X1; : : : ; X5) of length 5. We generated the data as follows: X1 Bernoulli(1=2) and Xj jX1 ; : : : ; Xj 1 It follows that f (x1 ; : : : ; x5 ) = = 1=4 if Xj 1 = 0 3=4 if Xj 1 = 1: x1 1 x1 x2 1 x2 x3 1 x3 x4 1 x4 3 3 3 3 3 3 3 1 3 2 4 2 4 4 4 4 x1+x2+x3+x4 4 x1 x2 x3 x4 1 3 3 4 4 : We estimated f using three methods: (i) maximum likelihood treating this as a multinomial with 32 categories, (ii) maximum likelihood from the best loglinear model using AIC and forward selection and (iii) maximum likelihood from the best loglinear 4 4 4 324 18. Loglinear Models model using BIC and forward selection. We estimated the risk by simulating the example 100 times. The average risks were: Method Risk MLE 0.63 AIC 0.54 BIC 0.53 In this example, there is little dierence between AIC and BIC. Both are better than maximum likelihood. 18.7 Bibliographic Remarks A classic references on loglinear models is Bishop, Fienberg and Holland (1975). See also Whittaker (1990) from which some of the exercises are borrowed. 18.8 Exercises 1. Solve for the p0ij s in terms of the 's in Example 18.3. 2. Repeat example 18.17 using 7 covariates and n = 1000. To avoid numerical problems, replace any zero count with a one. 3. Prove Lemma 18.5. 4. Prove Lemma 18.9. 5. Consider random variables (X1; X2; X3 ; X4). Suppose the log-density is log f (x) = ; (x) + 12 (x) + 13 (x) + 24 (x) + 34 (x): (a) Draw the graph G for these variables. (b) Write down all independence and conditional independence relations implied by the graph. 18.8 Exercises 325 (c) Is this model graphical? Is it hierarchical? 6. Suppose that parameters p(x1; x2 ; x3) are proportional to the following values: x1 0 0 0 1 0 2 8 1 16 128 x2 x3 1 1 0 1 4 16 32 256 Find the -terms for the log-linear expansion. Comment on the model. 7. Let X1; : : : ; X4 be binary. Draw the independence graphs corresponding to the following log-linear models. Also, identify whether each is graphical and/or hierarchical (or neither). (a) log f = 7 + 11x1 + 2x2 + 1:5x3 + 17x4 (b) log f = 7+11x1 +2x2 +1:5x3 +17x4 +12x2 x3 +78x2 x4 + 3x3 x4 + 32x2x3 x4 (c) log f = 7+11x1 +2x2 +1:5x3 +17x4 +12x2x3 +3x3x4 + x1 x4 + 2x1 x2 (d) log f = 7 + 5055x1x2 x3 x4 This is page 327 Printer: Opaque this 19 Causal Inference In this Chapter we discuss causation. Roughly speaking, the statement \X causes Y " means that changing the value of X will change the distribution of Y . When X causes Y , X and Y will be associated but the reverse is not, in general, true. Association does not necessarily imply causation. We will consider two frameworks for discussing causation. The rst uses the notation of counterfactual random variables. The second, presented in the next Chapter, uses directed acyclic graphs. 19.1 The Counterfactual Model Suppose that X is a binary treatment variable where X = 1 means \treated" and where X = 0 means \not treated." We are using the word \treatment" in a very broad sense: treatment might refer to a medication or something like smoking. An alternative to \treated/not treated" is \exposed/not exposed" but we shall use the former. 328 19. Causal Inference Let Y be some outcome variable such as presence or absence of disease. To distinguish the statement \X is associated Y " from the statement \X causes Y " we need to enrich our probabilistic vocabulary. We will decompose the response Y into a more negrained object. We introduce two new random variables (C0 ; C1 ), called potential outcomes with the following interpretation: C0 is the outcome if the subject is not treated (X = 0) and C1 is the outcome if the subject is treated (X = 1). Hence, Y= C0 if X = 0 C1 if X = 1: We can express the relationship between Y and (C0 ; C1 ) more succintly by Y = CX : (19.1) This equation is called the consistency relationship. Here is a toy data set to make the idea clear: X Y C0 C1 0 0 0 0 1 1 1 1 4 7 2 8 3 5 8 9 4 7 2 8 * * * * * * * * 3 5 8 9 The asterisks denote unobserved values. When X = 0 we don't observe C1 in which case we say that C1 is a counterfactual since it is the outcome you would have had if, counter to the fact, you had been treated (X = 1). Similarly, when X = 1 we don't observe C0 and we say that C0 is counterfactual. Notice that there are four types of subjects: 19.1 The Counterfactual Model 329 Type C0 C1 Survivors 1 1 Responders 0 1 Anti-repsonders 1 0 Doomed 0 0 Think of the potential outcomes (C0 ; C1 ) as hidden variables that contain all the relavent information about the subject. Dene the average causal eect or average treatment eect to be = E (C1 ) E (C0 ): (19.2) The parameter has the following interpretation: is the mean if everyone were treated (X = 1) minus the mean if everyone were not treated (X = 0). There are other ways of measuring the causal eect. For example, if C0 and C1 are binary, we dene the causal odds ratio P(C1 = 1) P(C1 = 0) 1) PP((CC0 == 0) 0 and the causal relative risk P(C1 = 1) : P(C0 = 1) The main ideas will be the same whatever causal eect we use. For simplicity, we shall work with the average causal eect . Dene the association to be = E (Y jX = 1) E (Y jX = 0): (19.3) Again, we could use odds ratios or other summaries if we wish. 19.1 (Association is not equal to Causation) In general, 6= . Theorem Example 19.2 Suppose the whole population is as follows: 330 19. Causal Inference X Y C0 C1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 Again, the asterisks denote unobserved values. Notice that C0 = C1 for every subject, thus, this treatment has no eect. Indeed, 8 8 1X 1X = E (C1 ) E (C0 ) = C C 8 i=1 1i 8 i=1 0i 1+1+1+1+0+0+0+0 1+1+1+1+0+0+0+0 = 8 8 = 0: Thus the average causal eect is 0. The observed data are X 's and Y 's only from which we can estimate the association: jX = 1) E (Y jX = 0) 1+1+1+1 0+0+0+0 = 1: = 4 4 = E (Y Hence, 6= . To add some intutition to this example, imagine that the outcome variable is 1 if \healthy" and 0 if \sick". Suppose that X = 0 means that the subject does not take vitamin C and that X = 1 means that the subject does take vitamin C. Vitamin C has no causal eect since C0 = C1 . In this example there are two types of people: healthy people (C0 ; C1 ) = (1; 1) and unhealthy people (C0 ; C1 ) = (0; 0). Healthy people tend to take vitamin C while unhealty people don't. It is this association between (C0 ; C1 ) and X that creates an association between X and Y . If we only had data on X and Y we would conclude that X 19.1 The Counterfactual Model 331 and Y are associated. Suppose we wrongly interpret this causally and conclude that vitamin C prevents illness. Next we might encourage everyone to take vitamin C. If most people comply the population will look something like this: X Y C0 C1 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 Now = (4=7) (0=1) = 4=7. We see that went down from 1 to 4/7. Of course, the causal eect never changed but the naive observer who does not distinguish association and causation will be confused because his advice seems to have made things worse instead of better. In the last example, = 0 and = 1. It is not hard to create examples in which > 0 and yet < 0. The fact that the association and causal eects can have dierent signs is very confusing to many people including well educated statisticians. The example makes it clear that, in general, we cannot use the association to estimate the causal eect . The reason that 6= is that (C0 ; C1 ) was not independent of X . That is, treatment assigment was not independent of person type. Can we ever estimate the causal eect? The answer is: sometimes. In particular, random asigment to treatment makes it possible to estimate . 19.3 Suppose we randomly assign subjects to treatment and that P(X = 0) > 0 and P(X = 1) > 0. Then = . Hence, any consistent estimator of is a consistent estimator Theorem 332 19. Causal Inference of . In particular, a consistent estimator is b = Eb (Y = Y1 jX = 1) Eb (Y jX = 0) Y0 is a consistent estimator of , where Y1 = 1 X n1 i:X =1 Yi ; Y0 = i n1 = 1 X Y; n0 i:X =0 i i Pn X and n = Pn (1 X ). 0 i i=1 i i=1 Since X is randomly assigned, X is independent of (C0 ; C1 ). Hence, Proof. = E (C1 ) E (C0 ) = E (C1 jX = 1) E (C0 jX = 0) = E (Y jX = 1) E (Y jX = 0) = : since X q (C0 ; C1 ) since Y = CX The consistency follows from the law of large numbers. If Z is a covariate, we dene the conditional causal eect by z = E (C1 jZ = z ) E (C0 jZ = z): For example, if Z denotes gender with values Z = 0 (women) and Z = 1 (men), then 0 is the causal eect among women and 1 is the causal eect among men. In a randomized experiment, z = E (Y jX = 1; Z = z ) E (Y jX = 0; Z = z ) and we can estimate the conditional causal eect using appropriate sample averages. 19.2 Beyond Binary Treatments 333 Summary Random variables: (C0 ; C1 ; X; Y ). Consistency relationship: Y = CX . Causal Eect: = E (C1 ) E (C0 ). Association: = E (Y jX = 1) E (Y jX = 0). Random assigment =) (C0 ; C1 ) q X =) = . 19.2 Beyond Binary Treatments Let us now generalize beyond the binary case. Suppose that X 2 X . For example, X could be the dose of a drug in which case X 2 R . The counterfactual vector (C0 ; C1 ) now becomes the counterfactual process fC (x) : x 2 Xg (19.4) where C (x) is the outcome a subject would have if he received dose x. The observed reponse is given by the consistency relation Y C (X ): (19.5) See Figure 19.1. The causal regression function is (x) = E (C (x)): (19.6) The regression function, which measures association, is r(x) = E (Y jX = x). 19.4 In general, (x) 6= r(x). However, when X is randomly assigned, (x) = r(x). Theorem An example in which (x) is constant but r(x) is not constant is shown in Figure 19.2. The gure shows the counterfactuals processes for four subjects. The dots represent Example 19.5 334 19. Causal Inference C (x) 7 6 5 Y = C (X ) 4 3 2 1 0 0 1 2 3 4 5 6 X 7 8 9 10 FIGURE 19.1. A counterfactual process C (x). The outcome Y is the value of the curve C (x) evaluated at the observed dose X . 11 x 19.3 Observational Studies and Confounding 335 their X values X1 ; X2 ; X3 ; X4 . Since Ci(x) is constant over x for all i, there is no causal eect and hence C (x) + C2 (x) + C3 (x) + C4 (x) (x) = 1 4 is constant. Changing the dose x will not change anyone's outcome. The four dots in the lower plot represent the observed data points Y1 = C1 (X1 ); Y2 = C2 (X2 ); Y3 = C3 (X3 ); Y4 = C4 (X4 ). The dotted line represents the regression r(x) = E (Y jX = x). Although there is no causal eect, there is an association since the regression curve r(x) is not constant. 19.3 Observational Studies and Confounding A study in which treatment (or exposure) is not randomly assigned, is called an observational study. In these studies, subjects select their own value of the exposure X . Many of the health studies your read about in the newspaper are like this. As we saw, association and causation could in general be quite dierent. This descrepancy occurs in non-randomized studies because the potential outcome C is not independent of treatment X . However, suppose we could nd groupings of subjects such that, within groups, X and fC (x) : x 2 Xg are independent. This would happen if the subjects are very similar within groups. For example, suppose we nd people who are very similar in age, gender, educational background and ethnic background. Among these people we might feel it is reasonable to assume that the choice of X is essentially random. These other variables are called confounding variables. If we denote these other variables collectively as Z , then we can express this idea by saying that fC (x) : x 2 Xg q X jZ: (19.7) Equation (19.7) means that, within groups of Z , the choice of treatment X does not depend on type, as represented by fC (x) : A more precise definition of confounding is given in the next Chapter. 336 19. Causal Inference 3 b C1 (x) b C2 (x) 2 b C3 (x) 1 b C4 (x) 0 0 1 2 3 4 b 3 b 2 Y3 b 1 Y4 b 0 0 1 2 3 5 x Y1 Y2 (x) 4 5 FIGURE 19.2. The top plot shows the counterfactual process C (x) for four subjects. The dots represent their X values. Since Ci (x) is constant over x for all i, there is no causal eect. Changing the dose will not change anyone's outcome. The lower plot shows the causal regression function (x) = (C1 (x) + C2 (x) + C3 (x) + C4 (x))=4. The four dots represent the observed data points Y1 = C1 (X1 ); Y2 = C2 (X2 ); Y3 = C3 (X3 ); Y4 = C4 (X4 ). The dotted line represents the regression r(x) = E (Y jX = x). There is no causal eect since Ci (x) is constant for all i. But there is an association since the regression curve r(x) is not constant. 19.3 Observational Studies and Confounding 337 x 2 Xg. If (19.7) holds and we observe Z then we say that there is no unmeasured confounding. Theorem 19.6 Suppose that (19.7) holds. Then, (x) = Z E (Y jX = x; Z = z)dFZ (z)dz: (19.8) If rb(x; z ) is a consistent estimate of the regression function E (Y jX = x; Z = z ), then a consistent estimate of (x) is n b(x) = 1 X rb(x; Zi): n i=1 In particular, if r(x; z ) = 0 + 1 x + 2 z is linear, then a consistent estimate of (x) is b(x) = b0 + b1 x + b2 Z n (19.9) where (b0 ; b1 ; b2 ) are the least squares estimators. It is useful to compare equation R(19.8) to E (Y jX = x) which can be written as E (Y jX = x) = E (Y jX = x; Z = z )dFZ jX (z jx). Remark 19.7 Epidemiologists call (19.8) the adjusted treatment eect. The process of computing adjusted treatment eects is called adjusting (or controlling) for confounding. The selection of what confounders Z to measure and control for requires scientic insight. Even after adjusting for confounders, we cannot be sure that there are not other confounding variables that we missed. This is why observational studies must be treated with healthy skepticism. Results from observational studies start to become believable when: (i) the results are replicated in many studies, (ii) each of the studies controlled for plausible confounding variables, (iii) there is a plausible scientic explanation for the existence of a causal relationship. 338 19. Causal Inference A good example is smoking and cancer. Numerous studies have shown a relationship between smoking and cancer even after adjusting for many confouding variables. Moreover, in laboratory studies, smoking has been shown to damage lung cells. Finally, a causal link between smoking and cancer has been found in randomized animal studies. It is this collection of evidence over many years that makes this a convincing case. One single observational study is not, by itself, strong evidence. Remember that when you read the newspaper. 19.4 Simpson's Paradox Simpson's paradox is a puzzling phenomenon that is dicussed in most statistics texts. Unfortunately, it is explained incorrectly in most statistics texts. The reason is that it is nearly impossible to explain the paradox without using counterfactuals (or directed acyclic graphs). Let X be a binary treatment variable, Y a binary outcome and Z a third binary variable such as gender. Suppose the joint distribution of X; Y; Z is X=1 X=0 Y =1 Y =0 .1500 .2250 .0375 .0875 Z = 1 (men) Y =1 Y =0 .1000 .0250 .2625 .1125 Z = 0 (women) The marginal distribution for (X; Y ) is 19.4 Simpson's Paradox X=1 X=0 339 Y =1 Y =0 .25 .30 .55 .25 .50 .20 .50 .45 1 Now, j P(Y = 1 X = 1) j P(Y = 1 X = 0) = P(Y = 1; X = 1) P(X = 1) :25 :30 :50 :50 = 0:1: P(Y = 1; X = 0) P(X = 0) = We might naively interpret this to mean that the treatment is bad for you since P(Y = 1jX = 1) < P(Y = 1jX = 0). Furthermore, among men, j P(Y = 1 X = 1; Z = 1) j P(Y = 1 X = 0; Z = 1) P(Y = 1; X = 1; Z = 1) = P(X = 1; Z = 1) = :15 :3750 = 0:1: :0375 :1250 P(Y = 1; X = 0; Z = 1) P(X = 0; Z = 1) Among women, j P(Y = 1 X = 1; Z = 0) j P(Y = 1 X = 0; Z = 0) P(Y = 1; X = 1; Z = 0) = P(X = 1; Z = 0) = :1 :1250 = 0:1: :2625 :3750 P(Y = 1; X = 0; Z = 0) P(X = 0; Z = 0) To summarize, we seem to have the following information: 340 19. Causal Inference Mathematical Statement P(Y = 1jX = 1) < P(Y = 1jX = 0) P(Y = 1jX = 1; Z = 1) > P(Y = 1jX = 0; Z = 1) P(Y = 1jX = 1; Z = 0) > P(Y = 1jX = 0; Z = 0) English Statement? treatment is harmful treatment is benecial to men treatment is benecial to women Clearly, something is amiss. There can't be a treatment which is good for men, good for women but bad overall. This is nonsense. The problem is with the set of English statements in the table. Our translation from math into english is specious. The inequality P(Y = 1jX = 1) < P(Y = 1jX = 0) does not mean that treatment is harmful. The phrase \treatment is harmful" should be written mathematically as P(C1 = 1) < P(C0 = 1). The phrase \treatment is harmful for men" should be written P(C1 = 1jZ = 1) < P(C0 = 1jZ = 1). The three mathematical statements in the table are not at all contradictory. It is only the translation into English that is wrong. Let us now show that a real Simpson's paradox cannot happen, that is, there cannot be a treatment that is benecial for men and women but harmful overall. Suppose that treatment is benecial for both sexes. Then j P(C1 = 1 Z = z ) > P(C0 = 1jZ = z ) for all z . It then follows that P(C1 = 1) = X z X j P(C1 = 1 Z = z )P(Z = z ) P(C0 = 1jZ = z )P(Z = z ) z = P(C0 = 1): > Hence, P(C1 = 1) > P(C0 = 1) so treatment is benecial overall. No paradox. 19.5 Bibliographic Remarks 19.5 341 Bibliographic Remarks The use of potential outcomes to clarify causation is due mainly to Jerzy Neyman and Don Rubin. Later developments are due to Jamie Robins, Paul Rosenbaum and others. A parallel development took place in econometrics by various people including Jim Heckman and Charles Manski. Currently, there are no textbook treatments of causal inference from the potential outcomes viewpoint. 19.6 Exercises 1. Create an example like Example 19.2 in which > 0 and < 0. 2. Prove Theorem 19.4. 3. Suppose you are given data (X1 ; Y1 ); : : : ; (Xn; Yn) from an observational study, where Xi 2 f0; 1g and Yi 2 f0; 1g. Although it is not possible to estimate the causal eect , it is possible to put bounds on . Find upper and lower bounds on that can be consistently estimated from the data. Show that the bounds have width 1. Hint: Note that E (C1 ) = E (C1 jX = 1)P(X = 1) + E (C1 jX = 0)P(X = 0). 4. Suppose that X 2 R and that, for each subject i, Ci (x) = 1i x. Each subject has their own slope 1i . Construct a joint distribution on (1 ; X ) such that P(1 > 0) = 1 but E (Y jX = x) is a decreasing function of x, where Y = C (X ). Interpret. Hint: Write f (1 ; x) = f (1 )f (xj1 ). Choose f (xj1 ) so that when 1 is large, x is small and when 1 is small x is large. This is page 343 Printer: Opaque this 20 Directed Graphs 20.1 Introduction Directed graphs are similar to undirected graphs except that there are arrows between vertices instead of edges. Like undirected graphs, directed graphs can be used to represent independence relations. They can also be used as an alternative to counterfactuals to represent causal relationships. Some people use the phrase Bayesian network to refer to a directed graph endowed with a probability distribution. This is a poor choice of terminology. Statistical inference for directed graphs can be performed using frequentist or Bayesian methods so it is misleading to call them Bayesian networks. 20.2 DAG's A directed graph G consists of a set of vertices V and an edge set E of ordered pairs of variables. If (X; Y ) 2 E then there is an arrow pointing from X to Y . See Figure 20.1. 344 20. Directed Graphs Y X Z FIGURE 20.1. A directed graph with E = f(Y; X ); (Y ; Z )g. V = fX; Y ; Z g and If an arrow connects two variables X and Y (in either direction) we say that X and Y are adjacent. If there is an arrow from X to Y then X is a parent of Y and Y is a child of X . The set of all parents of X is denoted by or (X ). A directed path from X to Y is a set of vertices beginning with X , ending with Y such that each pair is connected by an arrow and all the arrows point in the same direction as follows: X X ! ! Y or X Y A sequence of adjacent vertices staring with X and ending with Y but ignoring the direction of the arrows is called an undirected path. The sequence fX; Y; Z g in Figure 20.1 is an undirected path. X is an ancestor of Y is there is a directed path from X to Y . We also say that Y is a descendant of X . A conguration of the form: X !Y Z is called a collider. A conguration not of that form is called a non-collider, for example, X or X !Y !Z Y !Z 20.3 Probability and DAG's 345 A directed path that starts and ends at the same variable is called a cycle. A directed graph is acyclic if it has no cycles. In this case we say that the graph is a directed acyclic graph or DAG. From now on, we only deal with graphs that are DAG's. 20.3 Probability and DAG's Let G be a DAG with vertices V = (X1 ; : : : ; X ). k Denition 20.1 If P is a distribution for V with probability function p, we say that If P is represents P if p(v ) = Y k Markov to p(x i G or that G j ) (20.1) i i=1 where are the parents of X . The set of distributions represented by G is denoted by M (G ). i i Example 20.2 For the DAG in Figure 20.2, P only if its probability function p has the form 2 M (G ) if and p(x; y; z; w) = p(x)p(y )p(z j x; y )p(w j z ): The following theorem says that P 2 M (G ) if and only if the Markov Condition holds. Roughly speaking, the Markov Condition says that every variable W is independent of the \past" given its parents. Theorem 20.3 A distribution P lowing Markov Condition 2 M (G ) if and only if the fol- holds: for every variable W , fj W qW W (20.2) f denotes all the other variables except the parents and where W descendants of W . 346 20. Directed Graphs X Z W Y FIGURE 20.2. Another DAG. B A D E C FIGURE 20.3. Yet another DAG. Example 20.4 In Figure 20.2, the Markov Condition implies that X qY and W q fX; Y g j Z: Example 20.5 Consider the DAG in Figure 20.3. In this case probability function must factor like p(a; b; c; d; e) = p(a)p(bja)p(cja)p(djb; c)p(ejd): The Markov Condition implies the following independence relations: D q A j fB; C g; E q fA; B; C g j D and B q C j A X2 20.4 More Independence Relations 347 X3 X1 X5 X4 FIGURE 20.4. And yet another DAG. 20.4 More Independence Relations With undirected graphs, we saw that pairwise independence relations implied other independence relations. Luckily, we were able to deduce these other relations from the graph. The situation is similar for DAG's. The Markov Condition allows us to list some independence relations. These relations may logically imply other other independence relations. Consider the DAG in Figure 20.4. The Markov Condition implies: X1 q X2 ; X2 q fX1 ; X4 g; X4 q fX2 ; X3 g j X1 ; X3 q X4 j fX1 ; X2 g; X5 q fX1 ; X2 g j fX3 ; X4 g It turns out (but it is not obvious) that these conditions imply that fX ; X g q X j fX ; X g: 4 5 2 1 3 How do we nd these extra independence relations? The answer is \d-separation." d-separation can be summarized by three rules. Consider the four DAG's in Figure 20.5 and the DAG in Figure 20.6. The rst 3 DAG's in Figure 20.5 are non-colliders. The DAG in the lower right of Figure 20.5 is a collider. The DAG in Figure 20.6 is a collider with a descendent. In what follows, we implicitly assume that P is faithful to G which means that P has no extra independence relations other than those logically implied by the Markov Condition. 348 20. Directed Graphs X Y Z X Y Z X Y Z X Y Z FIGURE 20.5. The rst three DAG's have no colliders. The fourth DAG in the lower right corner has a collider at Y . X Y Z W FIGURE 20.6. A collider with a descendent. The rules of d-separation. Consider the DAG's in Figures 20.5 and 20.6. Rule (1). In a non-collider, X and Z are d-connected, but they are d-separated given Y . Rule (2). If X and Z collide at Y then X and Z are d-separated but they are d-connected given Y . Rule (3). Conditioning on the descendant of a collider has the same eect as conditioning on the collider. Thus in Figure 20.6, X and Z are d-separated but they are d-connected given W . Here is a more formal denition of d-separation. Let X and Y be distinct vertices and let W be a set of vertices not containing X or Y . Then X and Y are d-separated given W if there 20.4 More Independence Relations X U S1 V W 349 Y S2 FIGURE 20.7. d-separation explained. exists no undirected path U between X and Y such that (i) every collider on U has a descendant in W and (ii) no other vertex on U is in W . If U; V and W are distinct sets of vertices and U and V are not empty, then U and V are d-separated given W if for every X 2 U and Y 2 V , X and Y are d-separated given W . Vertices that are not d-separated are said to be dconnected. Example 20.6 Consider the DAG in Figure 20.7. From the d- seperation rules we conclude that: X and Y are d-separated (given the empty set) X and Y are d-connected given fS1 ; S2 g X and Y are d-separated given fS1 ; S2 ; V g Theorem 20.7 (Spirtes, Glymour and Scheines) Let A, B and C be disjoint sets of vertices. Then A q B j C if and only if A and B are d-separated by C . Example 20.8 The fact that conditioning on a collider creates dependence might not seem intuitive. Here is a whimsical example from Jordan (2003) that makes this idea more palatable. Your friend appears to be late for a meeting with you. There are two explanations: she was abducted by aliens or you forgot to set your watch ahead one hour for daylight savings time. See Figure 20.8 Aliens and Watch are blocked by a collider which implies they are marginally independent. This seems reasonable since, before we know anything about your friend being late, we would expect these variables to be independent. We would also expect that P(Aliens = yesjLate = Y es) > P(Aliens = yes); 350 20. Directed Graphs aliens watch late FIGURE 20.8. Jordan's alien example. Was your friend kidnapped by aliens or did you forget to set your watch? learning that your friend is late certainly increases the probability that she was abducted. But when we learn that you forgot to set your watch properly, we would lower the chance that your friend was abducted. Hence, P(Aliens = yesjLate = Y es) 6= P(Aliens = yesjLate = Y es; W atch = no). Thus, Aliens and Watch are dependent given Late. Graphs that look dierent may actually imply the same independence relations. If G is a DAG, we let I (G ) denote all the independence statements implied by G . Two DAG's G1 and G2 for the same variables V are Markov equivalent if I (G1 ) = I (G2 ). Given a DAG G , let skeleton(G ) denote the undirected graph obtained by replacing the arrows with undirected edges. and G2 are Markov equivalent if and only if (i) skeleton(G1 ) = skeleton(G2 ) and (ii) G1 and G2 have the same colliders. Theorem 20.9 Two DAG's G 1 Example 20.10 The rst three DAG's in Figure 20.5 are Markov equivalent. The DAG in the lower right of the Figure is not Markov equivalent to the others. 20.5 Estimation for DAG's Let G be a DAG. Assume that all the variables V = (X1 ; : : : ; X ) are discrete. The probability function can be written m p(v ) = Y k p(x i i=1 j ): i (20.3) 20.5 Estimation for DAG's 351 To estimate p(v ) we need to estimate p(x j ) for each i. Think of the parents of X as one discrete variable Xe with many levels. For example, suppose that X3 has parents X1 and X2 and that X1 2 f0; 1; 2g while X2 2 f0; 1g. We can regard the parents X1 and X2 as a single variable Xe3 dened by i i i Xe3 = i 8 1 > > > > > > <2 > > > > > : 3 4 5 6 if if if if if if Hence we can write p(v ) = X1 = 0; X2 = 0 X1 = 0; X2 = 1 X1 = 1; X2 = 0 X1 = 1; X2 = 1 X1 = 2; X2 = 0 X1 = 2; X2 = 1: Y k p(x j xe ): i (20.4) i i=1 Theorem 20.11 Let V1 ; : : : ; V be iid random vectors from disn tribution p given in (20.4). The maximum likelihood estimator of p is pb(v ) = Y k pb(x j xe ) i (20.5) i i=1 where #fi : X = x and Xe = xe g : #fi : Xe = xe g Example 20.12 Let V = (X; Y; Z ) have the DAG in the top left of Figure 20.5. Given n observations (X1 ; Y1; Z1 ); : : : ; (X ; Y ; Z ), the mle of p(x; y; z ) = p(x)p(y jx)p(z jy ) is #fi : X = xg ; pb(x) = pb(x j xe ) = i i i i i i i i n i n #fi : X = x and Y = y g pb(y jx) = ; #fi : Y = y g i i i and pb(z jy ) = #fi : Y = y and Z = z g : #fi : Z = z g i i i n n 352 20. Directed Graphs It is possible to extend these ideas to continuous random variables as well. For example, we might use some parametric model p(xj ; ) for each condiitonal density. The likelihood function is then x x L() = Y n p(V ; ) = YY n m i i=1 p(X j ; ) ij j j i=1 j =1 where X is the value of X for the ith data point and are the parameters for the j th conditional density. We can then proceed using maximum likelihood. So far, we have assumed that the structure of the DAG is given. One can also try to estimate the structure of the DAG itself from the data. For example, we could t every possible DAG using maximum likelihood and use AIC (or some other method) to choose a DAG. However, there are many possible DAG's so you would need much data for such a method to be reliable. Producing a valid, accurate condence set for the DAG structure would require astronomical sample sizes. DAG's are thus most useful for encoding conditional independence information rather than discovering it. ij 20.6 j j Causation Revisited We discussed causation in Chapter 19 using the idea of counterfactual random variables. A dierent approach to causation uses DAG's. The two approaches are mathematically equivalent though they appear to be quite dierent. In Chapter 19, the extra element we added to clarify causation was the idea of a counterfactual. In the DAG approach, the extra element is the idea of intervention. Consider the DAG in Figure 20.9. The probability function for a distribution consistent with this DAG has the form p(x; y; z ) = p(x)p(y jx)p(z jx; y ). Here is pseudo-code for generating from this distribution: 20.6 Causation Revisited X Y 353 Z FIGURE 20.9. Conditioning versus intervening. for i = 1; : : : ; n : x < p (x ) y < p j (y jx ) z < p j (z jx ; y ) i X i i Y X i Z X;Y i i i i i Suppose we repeat this code many times yielding data (x1 ; y1 ; z1 ); : : : ; (x ; y ; z ). Among all the times that we observe Y = y , how often is Z = z ? The answer to this question is given by the conditional distribution of Z jY . Specically, n P(Z = z jY = y) = P(Y = y; Z = z ) p(y; z ) = P(Y = y ) p(y ) P P p(x; y; z ) p(x) p(y jx) p(z jx; y ) = = p(y ) p(y ) X p(y jx) p(x) X p(x; y ) = p(z jx; y ) = p(z jx; y ) p(y ) p(y ) x X x = x p(z jx; y ) p(xjy ): x x Now suppose we intervene by changing the computer code. Specically, suppose we x Y at the value y . The code now looks like this: n n 354 20. Directed Graphs set Y = y for i = 1; : : : ; n x < p (x ) z < p j (z jx ; y ) i X i i Z X;Y i i Having set Y = y , how often was Z = z ? To answer, note that the intervention has changed the joint probability to be p (x; z ) = p(x)p(z jx; y ): The answer to our question is given by the marginal distribution X X p (z ) = p (x; z ) = p(x)p(z jx; y ): x x We shall denote this as P(Z = z jY := y ) or p(z jY := y ). We call P(Z = z jY = y ) conditioning by observation or passive conditioning. We call P(Z = z jY := y ) conditioning by intervention or active conditioning. Passive conditioning is used to answer a predictive question like: \Given that Joe smokes, what is the probability he will get lung cancer?" Active conditioning is used to answer a causal question like: \If Joe quits smoking, what is the probability he will get lung cancer?" Consider a pair (G ; P) where G is a DAG and P is a distribution for the variables V of the DAG. Let p denote the probability function for P. Consider intervening and xing a variable X to be equal to x. We represent the intervention by doing two things: (1) Create a new DAG G by removing all arrows pointing into X ; (2) Create a new distribution p (v ) = P(V = v jX := x) by removing the term p(xj ) from p(v ). X 20.6 Causation Revisited 355 The new pair (G ; P ) represents the intervention \set X = x." Example 20.13 You may have noticed a correlation between rain and having a wet lawn, that is, the variable \Rain" is not independent of the variable \Wet Lawn" and hence p (r; w) 6= p (r)p (w) where R denotes Rain and W denotes Wet Lawn. Consider the following two DAGS: R;W R W Rain ! Wet Lawn Rain Wet Lawn: The rst DAG implies that p(w; r) = p(r)p(wjr) while the second implies that p(w; r) = p(w)p(rjw) No matter what the joint distribution p(w; r) is, both graphs are correct. Both imply that R and W are not independent. But, intuitively, if we want a graph to indicate causation, the rst graph is right and the second is wrong. Throwing water on your lawn doesn't cause rain. The reason we feel the rst is correct while the second is wrong is because the interventions implied by the rst graph are correct. Look at the rst graph and form the intervention W = 1 where 1 denotes \wet lawn." Following the rules of intervention, we break the arrows into W to get the modied graph: Rain set Wet Lawn =1 with distribution p (r) = p(r). Thus P(R = r j W := w) = P(R = r ) tells us that \wet lawn" does not cause rain. Suppose we (wrongly) assume that the second graph is the correct causal graph and form the intervention W = 1 on the second graph. There are no arrows into W that need to be broken so the intervention graph is the same as the original graph. Thus p (r) = p(rjw) which would imply that changing \wet" changes \rain." Clearly, this is nonsense. Both are correct probability graphs but only the rst is correct causally. We know the correct causal graph by using background knowledge. 356 20. Directed Graphs Remark 20.14 We could try to learn the correct causal graph from data but this is dangerous. In fact it is impossible with two variables. With more than two variables there are methods that can nd the causal graph under certain assumptions but they are large sample methods and, furthermore, there is no way to ever know if the sample size you have is large enough to make the methods reliable. We can use DAG's to represent confouding variables. If X is a treatment and Y is an outcome, a confounding variable Z is a variable with arrows into both X and Y ; see Figure 20.10. It is easy to check, using the formalism of interventions, that the following facts are true. In a randomized study, the arrow betwen Z and X is broken. In this case, even with Z unobserved (represented by enclosing Z in a circle), the causal relationship between X and Y is estimable because it can be shown that E (Y jX := x) = E (Y jX = x) which does not involve the unobserved Z . In an observational study, with all confounders observed, we get R E (Y jX := x) = E (Y jX = x; Z = z )dF (z ) as in formula (19.8). If Z is unobserved then we R cannot estimate the causal eect because E (Y jX := x) = E (Y jX = x; Z = z )dF (z ) involves the unobserved Z . We can't just use X and Y since in this case. P(Y = y jX = x) 6= P(Y = y jX := x) which is just another way of saying that causation is not association. In fact, we can make a precise connection between DAG's and counterfactuals as follows. Suppose that X and Y are binary. Dene the confounding variable Z by Z Z 8 > > < 12 Z= 3 > : if if if 4 if (C0 ; C1 ) = (0; 0) (C0 ; C1 ) = (0; 1) (C0 ; C1 ) = (1; 0) (C0 ; C1 ) = (1; 1): 20.7 Bibliographic Remarks Z X Z Y X 357 Z Y X FIGURE 20.10. Randomized study; Observational study with measured confounders; Observational study with unmeasured confounders. The circled variables are unobserved. From this, you can make the correspondence between the DAG approach and the counterfactual approach explicit. I leave this for the interested reader. 20.7 Bibliographic Remarks There are a number of texts on DAG's including Edwards (1996) and Jordan (2003). The rst use of DAG's for representing causal relationships was by Wright (1934). Modern tratments are contained in Spirtes, Glymour, and Scheines (1990) and Pearl (2000). Robins, Scheines, Spirtes and Wasserman (2003) discuss the problems with estimating causal structure from data. 20.8 Exercises 1. Consider the three DAG's in Figure 20.5 without a collider. Prove that X q Z jY . 2. Consider the DAG in Figure 20.5 with a collider. Prove that X q Z and that X and Z are dependent given Y . 3. Let X 2 f0; 1g, Y 2 f0; 1g, Z 2 f0; 1; 2g. Suppose the distribution of (X; Y; Z ) is Markov to: X !Y !Z Y 358 20. Directed Graphs Create a joint distribution p(x; y; z ) that is Markov to this DAG. Generate 1000 random vectors from this distribution. Estimate the distribution from the data using maximum likelihood. Compare the estimated distribution to the true distribution. Let = (000 ; 001 ; : : : ; 112 ) where = P(X = r; Y = s; Z = t). Use the bootstrap to get standard errors and 95 per cent condence intervals for these 12 parameters. rst 4. Let V = (X; Y; Z ) have the following joint distribution X Y jX=x Z j X = x; Y = y Bernoulli 1 2 e 4x 2 Bernoulli e2( + ) 2 Bernoulli 1 + e2( + ) 1 + e 4x x 2 y x y 2 : (a) Find an expression for P(Z = z j Y = y ). In particular, nd P(Z = 1 j Y = 1). (b) Write a program to simulate the model. Conduct a simulation and compute P(Z = 1 j Y = 1) empirically. Plot this as a function of the simulation size N . It should converge to the theoretical value you computed in (a). (c) Write down an expression for P(Z = 1 j Y := y ). In particular, nd P(Z = 1 j Y := 1). 20.8 Exercises 359 (d) Modify your program to simulate the intervention \set Y = 1." Conduct a simulation and compute P(Z = 1 j Y := 1) empirically. Plot this as a function of the simulation size N . It should converge to the theoretical value you computed in (c). 5. This is a continuous, Gaussian version of the last question. Let V = (X; Y; Z ) have the following joint distribution X Y jX=x Z j X = x; Y = y Normal (0; 1) Normal (x; 1) Normal (y + x; 1): Here, ; and are xed parameters. Economists refer to models like this as structural equation models. (a) Find R an explicit expression for f (z j y) and E (Z j Y = y ) = zf (z j y )dz . (b) Find an explicit expression for f (z j Y := y ) and then R nd E (Z j Y := y ) zf (z j Y := y )dy . Compare to (b). (c) Find the joint distribution of (Y; Z ). Find the correlation between Y and Z . (d) Suppose that X is not observed and we try to make causal conclusions from the marginal distribution of (Y; Z ). (Think of X as unobserved confounding variables.) In particular, suppose we declare that Y causes Z if 6= 0 and 360 20. Directed Graphs we declare that Y does not cause Z if = 0. Show that this will lead to erroneous conclusions. (e) Suppose we conduct a randomized experiment in which Y is randomly assigned. To be concrete, suppose that X Y Z j X = x; Y = y Normal(0; 1) Normal(; 1) Normal(y + x; 1): Show that the method in (d) now yields correct conclusions i.e. = 0 if and only if f (z j Y := y ) does not depend on y . This is page 359 Printer: Opaque this 21 Nonparametric Curve Estimation In this Chapter we discuss nonparametric estimation of probability density functions and regression functions which we refer to as curve estimation. In Chapter 8 we saw that it is possible to consistently estimate a cumulative distribution function F without making any assumptions about F . If we want to estimate a probability density function f (x) or a regression function r(x) = E (Y jX = x) the situation is dierent. We cannot estimate these functions consistently without making some smoothness assumptions. Correspondingly, we will perform some sort of smoothing operation on the data. A simple example of a density estimator is a histogram, which we discuss in detail in Section 21.2. To form a histogram estimator of a density f , we divide the real line to disjoint sets called bins. The histogram estimator is piecewise constant function where the height of the function is proportional to number of observations in each bin; see Figure 21.3. The number of bins is an example of a smoothing parameter. If we smooth too 360 21. Nonparametric Curve Estimation gb(x) This is a function of the data This is the point at which we are evaluating bg() FIGURE 21.1. A curve estimate gb is random because it is a function of the data. The point x at which we evaluate gb is not a random variable. much (large bins) we get a highly biased estimator while if we smooth too little (small bins) we get a highly variable estimator. Much of curve estimation is concerned with trying to optimally balance variance and bias. 21.1 The Bias-Variance Tradeo Let g denote an unknown function and let bgn denote an estimator of g. Bear in mind that bgn(x) is a random function evaluated at a point x. bgn is random because it depends on the data. Indeed, we could be more explicit and write bgn(x) = hx (X ; : : : ; Xn ) to show that bgn (x) is a function of the data X ; : : : ; Xn and that the function could be dierent for each x. See Figure 21.1. As a loss function, we will use the integrated squared error (ISE): Z L(g; bgn) = (g (u) bgn(u)) du: (21.1) 1 1 2 The risk or mean integrated squared error (MISE) is R(f; fb) = E L(g; bg) : (21.2) 21.2 Histograms 361 Lemma 21.1 The risk can be written as R(g; bgn) = where Z b (x) dx + 2 Z v (x) dx (21.3) b(x) = E (bgn (x)) g (x) (21.4) is the bias of gbn (x) at a xed x and v (x) = V (bgn(x)) = E (bgn(x) E (b gn(x)) 2 ) (21.5) is the variance of bgn (x) at a xed x. In summary, RISK = BIAS + VARIANCE: 2 (21.6) When the data are over-smoothed, the bias term is large and the variance is small. When the data are under-smoothed the opposite is true; see Figure 21.2. This is called the bias-variance trade-o. Minimizing risk corresponds to balancing bias and variance. 21.2 Histograms Let X ; : : : ; Xn be iid on [0; 1] with density f . The restriction to [0; 1] is not crucial; we can always rescale the data to be on this interval. Let m be an integer and dene bins 1 1 ; B = 1 ; 2 ; : : : ; B = m 1 ; 1 : (21.7) B = 0; m m m m m 1 2 Dene the binwidth h = 1=m, let j beRthe number of observations in Bj , let pbj = j =n and let pj = Bj f (u)du. 21. Nonparametric Curve Estimation 0.06 0.08 0.10 362 0.04 Risk 0.02 Variance 0.00 Bias squared Optimal Smoothing 0.1 0.2 0.3 0.4 more smoothing −−−> FIGURE 21.2. The Bias-Variance trade-o. The bias increases and the variance decreases with the amount of smoothing. The optimal amount of smoothing, indicated by the vertical line, minimizes the risk = bias2 + variance. 0.5 21.2 Histograms 363 The histogram estimator is dened by fbn (x) = 8 > pb1 =h > > < pb =h 2 . .. > > > : pb =h m x 2 B1 x 2 B2 ... x 2 Bm which we can write more succinctly as fbn (x) = n X pbj j =1 h I (x 2 Bj ): (21.8) R Bj To understand the motivation for this estimator, let pj = f (u)du and note that, for x 2 Bj and h small, R f (u)du f (x)h p b p fbn (x) = j j = Bj f (x) = f (x): h h h Example 21.2 Figure 21.3 shows three dierent histograms based on n = 1; 266 data points from an astronomical sky survey. Each data point represents the distance from us to a galaxy. The galaxies lie on a \pencilbeam" pointing directly from the Earth out into space. Because of the nite speed of light, looking at galaxies farther and farther away corresponds to looking back in time. Choosing the right number of bins involves nding a good tradeo between bias and variance. We shall see later that the top left histogram has too few bins resulting in oversmoothing and too much bias. The bottom left histogram has too many bins resulting in undersmoothing and too few bins. The top right histogram is just right. The histogram reveals the presence of clusters of galaxies. Seeing how the size and number of galaxy clusters varies with time, helps cosmologists understand the evolution of the universe. The mean and variance of fbn(x) are given in the following Theorem. 21. Nonparametric Curve Estimation 0 0 2 10 4 20 6 30 8 40 10 50 12 60 14 364 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.10 Just Right 0 −8 −10 −14 −12 20 40 cross validation score 60 −6 Oversmoothed 0.15 0.00 0.05 0.10 Undersmoothed 0.15 0.20 0 200 400 600 800 number of bins FIGURE 21.3. Three versions of a histogram for the astronomy data. The top left histogram has too few bins. The bottom left histogram has too many bins. The top right histogram is just right. The lower, right plot shows the estimated risk versus the number of bins. 1000 0.20 21.2 Histograms 365 21.3 Consider xed x and xed m, and let Bj be the bin containing x. Then, pj (1 pj ) pj and V (fbn (x)) = : (21.9) E (fbn (x)) = h nh2 Theorem Let's take a closer look at the bias-variance tradeo using equation (21.9) . Consider some x 2 Bj . For any other u 2 Bj , f (u) f (x) + (u x)f 0 (x) and so Z Z 0 pj = f (u)du f (x) + (u x)f (x) du Bj Bj = f (x)h + hf 0 (x) h j Therefore, the bias b(x) is p b(x) = E (fbn (x)) f (x) = j f (x) h f (x)h + hf 0 (x) h j x 1 = 2 If xej is the center of the bin, then Z b (x) dx 2 Bj Z Bj (f 0(xe Z 1 j )) 0 Bj h = (f 0(xej )) 12 : b (x)dx = 2 m Z X 2 x dx 2 x dx 3 b (x)dx h (f 0 (xe =1 1 h j 2 2 j =1 Bj m h2 X = 12 j 1 2 Z x : x : 2 f (x) (f 0(x))2 h j 2 Therefore, 1 2 h f 0 (x) h j 1 2 j )) 2 m X j =1 h (f 0(xej )) 12 h2 12 2 Z 1 0 3 (f 0(x)) dx: 2 366 21. Nonparametric Curve Estimation Note that this increases as a function of h. Now consider the variance. For h small, 1 pj 1, so v (x) pj nh 2 f (x)h + hf 0 (x) h j = nh2 (x) fnh 1 2 x where we have kept only the dominant term. So, Z 1 v (x)dx : 1 nh 0 Note that this decreases with h. Putting all this together, we get: Theorem R 21.4 Suppose that (f 0 (u))2 du < 1. Then Z 1 ( f 0 (u)) du + : 12 nh h2 R(fbn ; f ) 2 (21.10) The value h that minimizes (21.10) is h = 1 R n1=3 1=3 6 (f 0(u)) du 2 : (21.11) With this choice of binwidth, R(fbn ; f ) where C = (3=4)2=3 R C n2=3 1=3 (f 0(u)) du 2 (21.12) . Theorem 21.4 is quite revealing. We see that with an optimally chosen binwidth, the MISE decreases to 0 at rate n = . By comparison, most parametric estimators converge at rate 2 3 21.2 Histograms 367 n 1 . The slower rate of convergence is the price we pay for being nonparametric. The formula for the optimal binwidth h is of theoretical interest but it is not useful in practice since it depends on the unknown function f . A practical way to choose the binwidth is to estimate the risk function and minimize over h. Recall that the loss function, which we now write as a function of h, is L(h) = = Z (fbn(x) f (x)) dx 2 Z fbn (x) dx 2 Z 2 fbn (x)f (x)dx + Z f 2 (x) dx: The last term does not depend on the binwidth h so minimizing the risk is equivalent to minimizing the expected value of J (h) = Z fbn2 (x) dx Z 2 fbn(x)f (x)dx: We shall refer to E (J (h)) as the risk, although it diers from R the true risk by the constant term f (x) dx. 2 Denition 21.5 The cross-validation estimator of risk is Jb(h) = Z 2 fbn (x) dx n 2X fb i (Xi ) n i=1 ( ) (21.13) where fb( i) is the histogram estimator obtained after removing the ith observation. We refer to Jb(h) as the crossvalidation score or estimated risk. Theorem 21.6 The cross-validation estimator is nearly unbiased: E (Jb(x)) E (J (x)): 368 21. Nonparametric Curve Estimation In principle, we need to recompute the histogram n times to compute Jb(h). Moreover, this has to be done for all values of h. Fortunately, there is a shortcut formula. Theorem 21.7 The following identity holds: Jb(h) m n+1 X pb2j : (n 1) j=1 2 = (n 1)h (21.14) Example 21.8 We used cross-validation in the astronomy exam- ple. The cross-validation function is quite at near its minimum. Any m in the range of 73 to 310 is an approximate minimizer but the resulting histogram does not change much over this range. The histogram in the top right plot in Figure 21.3 was constructed using m = 73 bins. The bottom right plot shows the estimated risk, or more precisely, Ab, plotted versus the number of bins. Next we want some sort of condence set for f . Suppose fbn is a histogram with m bins and binwidth h = 1=m. We cannot realistically make condence statements about the ne details of the true density f . Instead, we shall make condence statements about f at the resolution of the histogram. To this end, dene pj for x 2 Bj h fe(x) = where pj = R Bj (21.15) f (u)du which is a \histogramized" version of f . 21.9 Let m = m(n) be the number of bins in the histogram Assume that m(n) ! 1 and m(n) log n=n ! 0 as n ! 1. Dene Theorem fbn . `(x) = u(x) = nq max q o2 fbn (x) c; 0 2 b fn (x) + c (21.16) 21.2 Histograms where c= Then, z=(2m) r 2 m : n 369 (21.17) lim P `(x) fe(x) u(x) for all x 1 : n!1 (21.18) Here is an outline of the proof. A rigorous proof requires fairly sophisticated tools. From the central limit thep pbj orem, pbj N (pj ; pj (1 pj )=n). By the delta method, p p N ( pj ; 1=(4n)). Moreover, it can be shown that the pbj 's are approximately independent. Therefore, Proof. p p 2 n pbj pp Z (21.19) j j where Z ; : : : ; Zm N (0; 1). Let A be the event that `(x) fe(x) u(x) for all x. So, 1 A = = = = Therefore, P(Ac ) = = = n `(x) fe(x) u(x) for all x np `(x) q nq o p u(x) for all x fe(x) q fbn (x) c q n fbn (x) max x fe(x) q e f (x) o q fbn (x) + c o for all x o c : r ! r q q p b p j j fe(x) > c = P max P max fbn (x) >c x j h h p p p p P max pbj j > c h = P max 2 n pbj j > 2c j j p p P max 2 n pbj > z=(2m) j j m X P max Zj > z=(2m) P Zj > z=(2m) j j =1 p p p j j p p j j p p hn 370 21. Nonparametric Curve Estimation = m X = : m j =1 Example 21.10 Figure 21.4 shows a 95 per cent condence enve- lope for the astronomy data. We see that even with over 1,000 data points, there is still substantial uncertainty. 21.3 Kernel Density Estimation Histograms are discontinuous. Kernel density estimators are smoother and they converge faster to the true density than histograms. Let X ; : : : ; Xn denote the observed data, a sample from f . In this chapter, a kernel is dened to be anyR smooth funcR tion K suchR that K (x) 0, K (x) dx = 1, xK (x)dx = 0 and K x K (x)dx > 0. Two examples of kernels are the 1 2 2 Epanechnikov kernel K (x) = 3 4 0 p p (1 x =5)= 5 jxj < 5 2 (21.20) otherwise and the Gaussian (Normal) kernel K (x) = (2) = 1 2 e x2 =2 . Denition 21.11 Given a kernel K and a positive number h, called the bandwidth, the kernel density estimator is dened to be fb(x) n X 1 = n h1 K x h Xi : i (21.21) =1 An example of a kernel density estimator is show in Figure 21.5. The kernel estimator eectively puts a smoothed out lump of mass of size 1=n over each data point Xi. The bandwidth h 371 0 10 20 30 40 50 21.3 Kernel Density Estimation 0.00 0.05 0.10 0.15 FIGURE 21.4. 95 per cent condence envelope for astronomy data using m = 73 bins. 0.20 21. Nonparametric Curve Estimation 0.00 0.05 0.10 0.15 372 −10 −5 0 FIGURE 21.5. A kernel density estimator fb. At each point x, fb(x) is the average of the kernels centered over the data points Xi . The data points are indicated by short vertical bars. 5 21.3 Kernel Density Estimation 373 controls the amount of smoothing. When h is close to 0, fbn consists of a set of spikes, one at each data point. The height of the spikes tends to innity as h ! 0. When h ! 1, fbn tends to a uniform density. Example 21.12 Figure 21.6 shows kernel density estimators for the astronomy data using for three dierent bandwidths. In each case we used a Gaussian kernel. The properly smoothed kernel density estimator in the top right panel shows similar structure as the histogram. However, it is easier to see the clusters in the kernel estimator. To construct a kernel density estimator, we need to choose a kernel K and a bandwidth h. It can be shown theoretically and empirically that the choice of K is not crucial. However, the choice of bandwidth h is very important. As with the histogram, we can make a theoretical statement about how the risk of the estimator depends on the bandwidth. Theorem 21.13 Under weak assumptions on f and K , R Z K (x)dx 1 00 b R(f; fn ) K h (f (x)) + 4 nh R where K = x K (x)dx. The optimal bandwidth is 4 2 4 2 2 (21.22) 2 c1 2=5 c12=5 c3 1=5 h = (21.23) n1=5 R R R where c1 = x2 K (x)dx, c2 = K (x)2 dx and c3 = (f 00 (x))2 dx. With this choice of bandwidth, c R(f; fbn ) 44=5 n for some constant c4 > 0. n Write Kh(x; X ) = h P 1 b i Kh (x; Xi ). Thus, E [fn (x)] Proof. K ((x X )=h) and fbn (x) = = E [Kh(x; X )] and V [fbn(x)] = 1 21. Nonparametric Curve Estimation 0 0 2 10 20 4 30 6 40 8 50 10 60 12 70 374 0.00 0.05 0.10 0.15 0.20 0.00 0.05 0.15 0.20 just right −7.4 −7.6 0 −7.8 200 400 600 800 cross−validation score 1000 −7.2 1200 −7.0 1400 oversmoothed 0.10 0.00 0.05 0.10 0.15 0.20 0.002 0.004 0.006 0.008 bandwidth undersmoothed FIGURE 21.6. Kernel density estimators and estimated risk for the astronomy data. 0.010 0.012 21.3 Kernel Density Estimation n 1 V [Kh (x; X )]. E [Kh (x; X )] Now, = = Z Z Z 1 K x t f (t) dt h h K (u)f (u hu) du = K (u) f (u) hf (u) + 21 f 00 (u) + du Z 1 00 = f (u) + 2 h f (u) u K (u) du 0 2 R 375 2 R since K (x) dx = 1 and x K (x) dx = 0. The bias is 1 00 E [Kh (x; X )] f (x) k h f (x): 2 By a similar calculation, R f ( x ) K (x) dx : V [fb (x)] 2 2 2 n n hn The second result follows from integrating the squared bias plus the variance. We see that kernel estimators converge at rate n = while histograms converge at the slower rate n = . It can be shown that, under weak assumptions, there does not exist a nonparametric estimator that converges faster than n = . The expression for h depends on the unknown density f which makes the result of little practical use. As with the histograms, we shall use cross-validation to nd a bandwidth. Thus, we estimate the risk (up to a constant) by 4 5 2 3 4 5 Jb(h) = Z fb2 (x)dz 2 n1 n X i=1 fb i (Xi ) (21.24) where fb i is the kernel density estimator after omitting the i observation. th 376 21. Nonparametric Curve Estimation Theorem 21.14 For any h > 0, E h i Jb(h) = E [J (h)] : Also, Jb(h) hn1 XX 2 i j X K i h Xj 2 K (0) + nh (21.25) R where K (x) = K (2) (x) 2K (x) and K (2) (z ) = K (z y )K (y )dy . In particular, if K is a N(0,1) Gaussian kernel then K (2) (z ) is the N (0; 2) density. For large data sets, We then choose the bandwidth hn that minimizes Jb(h). A b f and (21.25) justication for this method is given by the following remarkable can be computed theorem due to Stone. quickly using the fast Fourier trans- Theorem 21.15 (Stone's Theorem.) Suppose that f is bounded. form. Let fbh denote the kernel estimator with bandwidth h and let hn denote the bandwidth chosen by cross-validation. Then, R 2 f (x) fbhn (x) inf h R f (x) dx fbh (x) 2 dx ! 1: P (21.26) Example 21.16 The top right panel of Figure 21.6 is based on cross-validation. These data are rounded which problems for cross-validation. Specically, it causes the minimizer to be h = 0. To overcome this problem, we added a small amount of random Normal noise to the data. The result is that Jb(h) is very smooth with a well dened minimum. Remark 21.17 Do not assume that, if the estimator fb is wiggly, then cross-validation has let you down. The eye is not a good judge of risk. 21.3 Kernel Density Estimation 377 To construct condence bands, we use something similar to histograms although the details are more complicated. The version described here is from Chaudhuri and Marron (1999). 378 21. Nonparametric Curve Estimation Condence Band for Kernel Density Estimator 1. Choose an evenly spaced grid of points V = fv ; : : : ; vN g. For every v 2 V , dene 1 v Xi Y (v ) = K : (21.27) 1 i h h Note that fbn(v) = Y n(v), the average of the Yi(v)0s. 2. Dene s(v ) pn P where s2(v) = (n 1) 1 ni=1(Yi(v) Y n(v))2. se (v ) = (21.28) 3. Compute the eective sample size ESS (v ) = Pn (v Xi ) i=1 K h K (0) : (21.29) Roughly, this is the number of data points being averaged together to form fbn(v). 4. Let V = fv 2 V : ESS (v) 5g. Now dene the number of independent blocks m by 1 = ESS m n where ESS is the average of ESS over V . 5. Let q= and dene 1 1 + (1 ) =m 2 1 (21.30) (21.31) `(v ) = fbn (v ) q se (v ) and u(v ) = fbn (v ) + q se (v ) (21.32) where we replace `(v) with 0 if it becomes negative. Then, P n `(v ) f (v ) u(v ) for all v o 1 : 21.3 Kernel Density Estimation 379 Example 21.18 Figure 21.7 shows approximate 95 per cent con- dence bands for the astronomy data. Suppose now that the data Xi = (Xi ; : : : ; Xid ) are d-dimensional. The kernel estimator can easily be generalized to d dimensions. Let h = (h ; : : : ; hd) be a vector of bandwidths and dene 1 1 fbn (x) n X 1 = n Kh(x Xi ) where Kh (x Xi ) = (21.33) i=1 1 nh1 hd ( d Y j =1 K xi Xij ) (21.34) hj where h ; : : : ; hd are bandwidths. For simplicity, we might take hj = sj h where sj is the standard deviation of the j variable. There is now only a single bandwidth h to choose. Using calculations like those in the one-dimensional case, the risk is given by 1 th 1 R(f; fbn ) K 4 4 " d X R j =1 h4j Z fjj2 (x)dx + d X j 6=k h2j h2k Z # fjj fkk dx K (x)dx + nh h d where fjj is the second partial derivative of f . The optimal bandwidth satises hi c n = d ) leading to a risk of order n = d . From this fact, we see that the risk increases quickly with dimension, a problem usually called the curse of dimensionality. To get a sense of how serious this problem is, consider the following table from Silverman (1986) which shows the sample size required to ensure a relative mean squared error less than 0.1 at 0 when the density is multivariate normal and the optimal bandwidth is selected. 2 1 1 4 (4+ ) 1 (4+ ) 21. Nonparametric Curve Estimation 0 10 20 30 40 380 0.00 0.05 0.10 0.15 FIGURE 21.7. 95 per cent Condence bands for kernel density estimate for the astronomy data. 0.20 21.4 Nonparametric Regression 381 Dimension Sample Size 1 4 2 19 3 67 4 223 5 768 6 2790 7 10,700 8 43,700 9 187,000 10 842,000 This is bad news indeed. It says that having 842,000 observations in a ten dimensional problem is really like having 4 observations. 21.4 Nonparametric Regression Consider pairs of points (x ; Y ); : : : ; (xn; Yn) related by 1 1 Yi = r(xi ) + i (21.35) where E (i ) = 0 and we are treating the xi 's as xed. In nonparametric regression, we want to estimate the regression function r(x) = E (Y jX = x). There are many nonparametric regression estimators. Most involve estimating r(x) by taking some sort of weighted average of the Yi's, giving higher weight to those points near x. In particular, the Nadaraya-Watson kernel estimator is dened by 382 21. Nonparametric Curve Estimation rb(x) = n X i=1 wi (x)Yi (21.36) where the weights wi(x) are given by wi (x) = x xi h Pn x xj : K j =1 h K (21.37) The form of this estimator comes from rst estimating the joint density f (x; y) using kernel density estimation and then inserting this into: R Z yf (x; y )dy r(x) = E (Y jX = x) = yf (y jx)dy = R : f (x; y )dy Theorem 21.19 Suppose that V (i ) = . The risk of the Nadaraya2 Watson kernel estimator is Z 4 Z h4 2 2 x K (x)dx R(rbn ; r) 4 + Z 2 R f 0 (x) 2 00 0 r (x) + 2r (x) dx f (x) K (x)dx dx: nhf (x) 2 (21.38) The optimal bandwidth decreases at rate n choice the risk decreases at rate n 4=5 . = 1 5 and with this In practice, to choose the bandwidth h we minimize the cross validation score Jb(h) = n X i=1 (Yi rb i(xi )) (21.39) 2 where rb i is the estimator we get by omitting the i variable. An approximation to Jb is given by n X 1 b (21.40) J (h) = (Yi rb(xi )) : K i 1 Pn K xi xj th 2 2 (0) =1 j =1 h 21.4 Nonparametric Regression 383 Example 21.20 Figures 21.8 shows cosmic microwave background (CMB) data from BOOMERaNG (Nettereld et al. 2001), Maxima (Lee et al. 2001) and DASI (Halverson 2001). The data consist of n pairs (x1 ; Y1 ); : : : ; (xn ; Yn) where xi is called the multipole moment and Yi is the estimated power spectrum of the temperature uctuations. What your seeing are sound waves in the cosmic microwave background radiation which is the heat, left over from the big bang. If r(x) denotes the true power spectrum then Yi = r(xi ) + i where i is a random error with mean 0. The location and size of peaks in r(x) provides valuable clues about the behavior of the early universe. Figure 21.8 shows the t based on crossvalidation as well as an undersmoothed and oversmoothed t. The cross-validation t shows the presence of three well dened peaks, as predicted by the physics of the big bang. The procedure for nding condence bands is similar to that for density estimation. However, we rst need to estimate . Suppose that the xi's are ordered. Assuming r(x) is smooth, we have r(xi ) r(xi) 0 and hence 2 +1 Yi+1 h Yi = r(xi+1 ) + i+1 i h r(xi ) + i i i i +1 and hence Yi ) V (i+1 V (Yi+1 i ) = V (i+1 ) + V (i ) = 2 2 : We can thus use the average of the n 1 dierences Yi to estimate . Hence, dene +1 2 b2 = 1 n 1 X 2(n 1) i (Yi =1 +1 Yi)2 : Yi (21.41) 5000 4000 3000 2000 1000 0 0 1000 2000 3000 4000 5000 6000 21. Nonparametric Curve Estimation 6000 384 200 400 600 800 1000 200 400 800 1000 Oversmoothed 0 6e+05 5e+05 3e+05 1000 4e+05 2000 3000 estimated risk 4000 5000 7e+05 6000 8e+05 Undersmoothed 600 200 400 600 800 1000 20 40 60 80 bandwidth Just Right (Using cross−valdiation) FIGURE 21.8. Regression analysis of the CMB data. The rst t is undersmoothed, the second is oversmoothed and the third is based on cross-validation. The last panel shows the estimated risk versus the bandwidth of the smoother. The data are from BOOMERaNG, Maxima and DASI. 100 120 21.4 Nonparametric Regression 385 Condence Bands for Kernel Regression Follow the same procedure as for kernel density estimators except, change the denition of se (v) to v u n uX se (v ) = bt w2 (xi ) (21.42) i=1 where b is dened in (21.41). Example 21.21 Figure 21.9 shows a 95 per cent condence enve- lope for the CMB data. We see that we are highly condence of the existence and position of the rst peak. We are more uncertain about the second and third peak. At the time of this writing, more accurate data are becoming available that apparently provide sharper estimates of the second and third peak. The extension to multiple regressors X = (X ; : : : ; Xp) is straightforward. As with kernel density estimation we just replace the kernel with a multivariate kernel. However, the same caveats about the curse of dimensionality apply. In some cases, we might consider putting some restrictions on the regression function which will then reduce the curse of dimensionality. For example, additive regression is based on the model 1 Y= p X j =1 rj (Xj ) + : (21.43) Now we only need to t p one-dimensional functions. The model can be enriched by adding various interactions, for example, Y= p X j =1 rj (Xj ) + X j<k rjk (Xj Xk ) + : (21.44) Additive models are usually t by an algorithm called backtting. 21. Nonparametric Curve Estimation −1000 0 1000 2000 3000 4000 5000 6000 386 200 400 600 800 FIGURE 21.9. 95 per cent condence envelope for the CMB data. 1000 21.5 Appendix: Condence Sets and Bias 387 Backtting 1. Initialize r (x ), . . . , rp(xp ). 1 1 2. For j = 1; : : : ; p: P (a) Let i = Yi s6 j rs (xi ). (b) Let rj be the function estimate obtained by regressing the i 's on the j covariate. = th 3. If converged STOP. Else, go back to step 2. Additive models have the advantage that they avoid the curse of dimensionality and they can be t quickly but they have one disadvantage: the model is not fully nonparametric. In other words, the true regression function r(x) may not be of the form (21.43). 21.5 Appendix: Condence Sets and Bias The condence bands we computed are not for the density function or regression function but rather, for the smoothed function. For example, the condence band for a kernel density estimate with bandwidth h is a band for the function one gets by smoothing the true function with a kernel with the same bandwidth. Getting a condence set for the true function is complicated for reasons we now explain. First, let's review the parametric case. When estimating a scalar quantity with an estimator b, the usual condence interval is of the form q bz= sn where b is the maximum likelihood estimator and sn = V (b) is the estimated standard error of the estimator. Under standard regularity conditions, b N (; sn) 2 388 21. Nonparametric Curve Estimation and lim P b 2 sn b + 2 sn = 1 : n!1 But let us take a closer look. Let bn = E (bn) We know that b N (; sn ) but a more accurate statement is b N ( + bn ; sn). We usually ignore the bias term bn in large sample calculations but let's keep track of it. The coverage of the usual condence interval is P b z=2 sn b + z=2 sn = z=2 P = P ! z=2 sn N (bn ; sn) z=2 z=2 sn bn ; 1 z=2 : z=2 N sn P = b In a well behaved parametric model, sn is of size n = bn is of size n . Hence, bn=sn ! 0 so the last displayed probability statement becomes P( z= N (0; 1) z= ) = 1 . What makes parametric condence intervals have the right coverage is the fact that bn=sn ! 0. The situation is more complicated for kernel methods. Consider estimating a density f (x) at a single point x with a kernel density estimator. Since fb(x) is a sum if iid random variables, the central limit theorem implies that c f (x) b f (x) N f (x) + b (x); (21.45) 1 2 1 2 2 2 n where nh 1 (21.46) 2 R R is the bias, c = x K (x)dx and c = K (x)dx. The estimated standard error is bn (x) = h2 f 00 (x)c1 1 2 2 2 ( c fb(x) sn (x) = 2 nh )1=2 : (21.47) 21.6 Bibliographic Remarks 389 Suppose we use the usual interval fb(x) z= sn. Arguing as above, the coverage is approximately, bn (x) P z= N ; 1 z= : sn (x) 2 2 2 The optimal bandwidth is of the form h = cn = for some constant c. If we plug h = cn = into (21.46) and(21.47) we see that b (x)=sn(x) does not tend to 0. Thus, the condence interval will have coverage less than 1 . 1 5 1 5 n 21.6 Bibliographic Remarks Two very good books on curve estimation are Scott (1992) and Silverman (1986). I drew heavily on those books for this Chapter. 21.7 Exercises 1. Let X ; : : : ; Xn f and let fbn be the kernel density estimator using the boxcar kernel: <x< K (x) = 01 otherwise : 1 1 2 1 2 (a) Show that E (fb(x)) and V (fb(x)) = 1 nh2 2 4 Z 1 = h Z x+(h=2) x (h=2) x+(h=2) x (h=2) f (y )dy f (y )dy Z x+(h=2) x (h=2) f (y )dy !2 3 5: 390 21. Nonparametric Curve Estimation (b) Show that if h ! 0 and nh ! 1 as n ! 1 then fbn (x) ! f (x). P 2. Get the data on fragments of glass collected in forensic work. You need to download the MASS library for R then: library(MASS) data(fgl) x <- fgl[[1]] help(fgl) The data are also on my website. Estimate the density of the rst variable (refractive index) using a histogram and use a kernel density estimator. Use cross-validation to choose the amount of smoothing. Experiment with different binwidths and bandwidths. Comment on the similarities and dierences. Construct 95 per cent condence bands for your estimators. 3. Consider the data from question 2. Let Y be refractive index and let x be aluminium content (the fourth variable). Do a nonparametric regression to t the model Y = f (x) + . Use cross-validation to estimate the bandwidth. Construct 95 per cent condence bands for your estimate. 4. Prove Lemma 21.1. 5. Prove Theorem 21.3. 6. Prove Theorem 21.7. 7. Prove Theorem 21.14. 21.7 Exercises 391 8. Consider regression data (x ; Y ); : : : ; (xn; Yn). Suppose that 0 xi 1 for all i. Dene bins Bj as in equation (21.7). For x 2 Bj dene rbn (x) = Y j where Y j is the mean of all the Yi's corresponding to those xi 's in Bj . Find the approximate risk of this estimator. From this expression for the risk, nd the optimal bandwidth. At what rate does the risk go to zero? 1 1 9. Show that with suitable smoothness assumptions on r(x), b in equation (21.41) is a consistent estimator of . 2 2 This is page 393 Printer: Opaque this 22 Smoothing Using Orthogonal Functions In this Chapter we study a dierent approach to nonparametric curve estimation based on orthogonal functions. We begin with a brief introduction to the theory of orthogonal functions. Then we turn to density estimation and regression. L2 Spaces Let v = (v ; v ; v ) denote a three dimensional vector, that is, a list of three real numbers. Let V denote the set of all such vectors. If a is a scalar (a number) and v is a vector, we dene av = (av ; av ; av ). The sum of vectors v and w is dened by v + w = (v + w ; v + w ; v + w ). The inner product between P two vectors v and w is dened by hv; wi = i viwi. The norm (or length) of a vector v is dened by 22.1 Orthogonal Functions and 1 1 2 3 2 1 3 1 2 2 3 p 3 jjvjj = hv; vi = 3 =1 v u 3 uX t vi2 : i=1 (22.1) 394 22. Smoothing Using Orthogonal Functions Two vectors are orthogonal (or perpendicular) if hv; wi = 0. A set of vectors are orthogonal if each pair in the set is orthogonal. A vector is normal if jjvjj = 1. Let = (1; 0; 0), = (0; 1; 0), = (0; 0; 1). These vectors are said to be an orthonormal basis for V since they have the following properties: (i) they are orthogonal; (ii) they are normal; (iii) they form a basis for V which means that if v 2 V then v can be written as a linear combination of , , : 1 2 3 1 v= X 3 j =1 where j j 2 3 j = hj ; v i: (22.2) For example, if v = (12; 3; 4) then v = 12 +3 +4 . There are other orthonormal bases for V , for example, 1 1 1 1 1 1 1 2 = p3 ; p3 ; p3 ; = p2 ; p2 ; 0 ; = p6 ; p6 ; p6 : You can check that these three vectors also form an orthonormal basis for V . Again, if v is any vector then we can write 1 2 2 1 v= 3 X j =1 j 3 3 where j j = h j ; v i: For example, if v = (12; 3; 4) then v = 10:97 + 6:36 + 2:86 : Now we make the leap from vectors to functions. Basically, we just replace vectors with functions and sums with integrals. Let L (a;Rb) denote all functions dened on the interval [a; b] such that ab f (x) dx < 1: 1 2 3 2 2 L2 [a; b] = f : [a; b] ! R ; Z b a f (x) dx < 1 : 2 (22.3) 22.1 Orthogonal Functions and L2 Spaces 395 We sometimes write L instead of L (a; b). RThe inner product between two functions f; g 2 L is dened by f (x)g(x)dx. The norm of f is sZ f (x) dx: (22.4) jjf jj = 2 2 2 2 R Two functions are orthogonal if f (x)g(x)dx = 0. A function is normal if jjf jj = 1. R A sequence of functions ; ; ; : : : is orthonormal if j (x)dx = R 1 for each j and i(x)j (x)dx = 0 for i 6= j . An orthonormal sequence is complete if the only function that is orthogonal to each j is the zero function. In this case, the functions ; ; ; : : : form in basis, meaning that if f 2 L then f can The the be written as 1 1 2 2 2 3 3 f (x) = equality in displayed equationR means that (f (x) 2 1 X j =1 j j (x); where Parseval's relation says that Z jjf jj f (x)dx = 2 2 where = ( ; ; : : :). 1 j = 1 X j =1 Z b a f (x)j (x)dx: (22.5) (22.6) j2 jj jj2 2 Example 22.1 An example of an orthonormal basis for L2 [0; 1] is the dened as follows. Let 1 (x) = 1 and for j 2 dene p cosine basis j (x) = 2 cos((j 1)x): The rst six functions are plotted in Figure 22.1. Example 22.2 Let 2 :1 f (x) = x(1 x) sin (x + :05) p (22.7) fn (x))2 dx ! where fn (x) Pn j =1 j j (x). 0 = 22. Smoothing Using Orthogonal Functions −1.5 0.6 −1.0 0.8 −0.5 1.0 0.0 0.5 1.2 1.0 1.4 1.5 396 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 −1.5 −1.5 −1.0 −1.0 −0.5 −0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 −1.5 −1.5 −1.0 −1.0 −0.5 −0.5 0.0 0.0 0.5 0.5 1.0 1.0 1.5 1.5 0.0 FIGURE 22.1. The rst six functions in the cosine basis. 22.1 Orthogonal Functions and L2 Spaces 397 which is called the \doppler function." Figure 22.2 shows f (top left) and its approximation fJ (x) = J X j =1 j j (x) with J equal to 5 (top right), 20 (bottom left) and 200 (bottom right). As J increases we see that fJ (x) gets closer to f (x). The R1 coeÆcients j = 0 f (x)j (x)dx were computed numerically. Example 22.3 The ne by Pj (x) = Legendre polynomials on [ 1; 1] are de- 1 dj 2 2j j ! dxj (x 1)j ; j = 0; 1; 2; : : : (22.8) It can be shown that these functions are complete and orthogonal and that Z 1 2 : (22.9) 2j + 1 p It follows that the functions j (x) = (2j + 1)=2Pj (x), j = 0; 1; : : : form an orthonormal basis for L ( 1; 1). The rst few 1 Pj2(x)dx = 2 Legendre polynomials are P0 (x) P1 (x) = = P (x) = P (x) = 2 3 1 x 1 3x 1 2 1 5x 3x: 2 2 3 These polynomials may be constructed explicitly using the following recursive relation: Pj +1 (x) = (2j + 1)xPj (x) j+1 jPj 1 (x) : (22.10) 22. Smoothing Using Orthogonal Functions −0.4 −0.1 −0.2 0.0 0.0 0.1 0.2 0.2 0.4 398 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.4 −0.2 −0.2 0.0 0.0 0.2 0.2 0.4 0.4 0.0 FIGURE 22.2. Approximating the doppler function p 2:1 f (x) = x(1 x) sin (x+:05) with its expansion in the cosine basis. The P function f (top left) and its approximation fJ (x) = Jj=1 j j (x) with J equal to 5 (top right), 20 (bottom R 1 left) and 200 (bottom right). The coeÆcients j = 0 f (x)j (x)dx were computed numerically. 22.2 Density Estimation 399 The coeÆcients ; ; : : : are related to the smoothness of the function f . To see why, note that if f is smooth, then its derivatives will be nite. Thus we expect that, for some k, R (f k (x)) dx < 1 where f k is ther k derivative of f . Now P1 consider the cosine basis (22.7) and let f (x) = j j j (x). Then, 1 1 ( ) 2 2 ( ) th 0 =1 Z 1 0 (f k (x)) dx = 2 ( ) 2 P1 2 j =1 j 1 X j =1 j2 ( (j 1)) k : 2 The only way that ((j 1)) k can be nite is if the j 's get small when j gets large. To summarize: 2 If the function f is smooth then the coecients j will be small when j is large. For the rest of this chapter, assume we are using the cosine basis unless otherwise specied. 22.2 Density Estimation Let X ; : : : ; Xn be observations from a distribution on [0; 1] with density f . Assuming f 2 L we can write iid 1 2 f (x) = 1 X j =1 j j (x) where ; ; : : : is an orthonormal basis. Dene n X 1 b = (X ): 1 2 j Theorem n i=1 j i (22.11) 22.4 The mean and variance of bj are E bj V bj = = j j2 n (22.12) (22.13) 400 22. Smoothing Using Orthogonal Functions where Z = V (j (Xi)) = (j (x) j ) f (x)dx: (22.14) We have n X 1 b E j = n E (j (Xi)) i = EZ (j (X )) = j (x)f (x)dx = j : The calculation for the variance is similar. Hence, bj isPan unbiased estimate of j . It is tempting to estimate f by 1j bj j (x) but this turns out to have a very high variance. Instead, consider the estimator j 2 2 Proof. =1 1 =1 ( )= fb x J X j =1 bj j (x): (22.15) The number of terms J is a smoothing parameter. Increasing J will decrease bias while increasing variance. For technical reasons, we restrict J to lie in the range 1J p p where p = p(n) = n. To emphasize the dependence of the risk function on J , we write the risk function as R(J ). Theorem 22.5 The risk of fb is given by R(J ) = J X j2 j =1 n + 1 X j =J +1 (22.16) j2 : In kernel estimation, we used cross-validation to estimate the risk. In the orthogonal function approach, we instead use the risk estimator Rb (J ) = J X bj2 j =1 n + p X j =J +1 bj2 bj2 n + (22.17) 22.2 Density Estimation 401 where a = maxfa; 0g and + n X 1 j (Xi ) bj = n 1i 2 (22.18) bj : 2 =1 To motivate this estimator, note that bj is an unbiased estmate of j and bj bj is an unbiased estimator of j . We take the positive part of the latter term since we know that j cannot be negative. We now choose 1 Jb p to minimize Rb(f;b f ). Here is a summary. 2 2 2 2 2 2 Summary of Orthogonal Function Density Estimation 1. Let bj n X 1 = (X ): n i=1 j i 2. Choose Jb to minimize Rb(J ) over 1 J p = pn where Rb(J ) = and J X bj2 j =1 + n p X j =J +1 bj 2 n X 1 bj = j (Xi ) n 1i 2 bj2 n + 2 bj : =1 3. Let ( )= fb x Jb X j =1 bj j (x): The estimator fbn can be negative. If we are interested in exploring the shape of f , this is not a problem. However, if we need our estimate to be a probability density function, we 402 22. Smoothing Using Orthogonal Functions can truncate the estimate and then normalize it. That, we take R fb = maxffbn (x); 0g= maxffbn(u); 0gdu. Now let us construct a condence band for f . Suppose we estimate f using JPorthogonal functions. We are essentially estimating fe(x) = Jj j j (x) not the true density f (x) = P1 j j j (x). Thus, the condence band should be regarded as a band for fe(x). Theorem 22.6 An approximate 1 condence band for fe is (`(x); u(x)) where `(x) = fbn (x) c; u(x) = fbn (x) + c (22.19) 1 0 =1 =1 where JK c= p n and p 1 + p2z s 2 J (22.20) K = 1max max jj (x)j: j J x For the cosine basis, K p = 2. Example 22.7 Let 5 1 X (x; ; :1) f (x) = (x; 0; 1) + j 6 6j 5 =1 where (x; ; ) denotes a Normal density with mean and standard deviation , and (1 ; : : : ; 5 ) = ( 1; 1=2; 0; 1=2; 1). Marron and Wand (1990) call this \the claw" although I prefer the \Bart Simpson." Figure 22.3 shows the true density as well as the estimated density based on n = 5; 000 observations and a 95 per cent condence band. The density has been rescaled to have most of its mass between 0 and 1 using the transformation y = (x + 3)=6. 403 0 1 2 3 4 22.2 Density Estimation 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0 1 2 3 4 0.0 FIGURE 22.3. The top plot is the true density for the Bart Simpson distribution (rescaled to have most of its mass between 0 and 1). The bottom plot is the orthogonal function density estimate and 95 per cent condence band. 404 22. Smoothing Using Orthogonal Functions 22.3 Regression Consider the regression model Yi = r(xi ) + i ; i = 1; : : : ; n (22.21) where the i are independent with mean 0 and variance . We will initially focus on the special case where xi = i=n. We assume that r 2 L [0; 1] and hence we can write 2 2 r(x) = 1 X j =1 j j (x) where Z j = 1 0 r(x)j (x)dx (22.22) where ; ; : : : where is an orthonormal basis for [0; 1]. Dene n 1X Yi j (xi ); j = 1; 2; : : : (22.23) bj = n 1 2 i=1 Since bj is an average, the central limit theorem tells us that bj will be approximately normally distributed. Theorem 22.8 2 bj N j ; : n (22.24) The mean of bj is n n X X 1 1 b E (Y ) (x ) = r(x ) (x ) E ( ) = Proof. j i n Z i=1 j i n r(x)j (x)dx = j i=1 i j i where the approximate equality follows from the denition of a R P Riemann integral: i n h(xi) ! h(x)dx where n = 1=n. The variance is n X 1 b V ( ) = V (Y ) (x ) 1 0 j n2 i=1 i 2 j i 22.3 Regression = 405 n n 2 X 2 1 X 2 ( x ) = 2 (x ) n2 i=1 j i n n i=1 j i Z 2 2 2 j (x)dx = n n since R j (x)dx = 1. As we did for density estimation, we will estimate r by 2 rb(x) = Let R(J ) = E Z J X j =1 bj j (x): (r(x) rb(x))2 dx be the risk of the estimator. P Theorem 22.9 The risk R(J ) of the estimator rbn (x) = Jj =1 is R(J ) = 1 J 2 X + 2: n j =J +1 j bj j (x) (22.25) To estimate for = V ar(i ) we use 2 b2 = n n X b2 k i=n k+1 j (22.26) where k = n=4. To motivate this estimator, recall that if f is smooth then j p0 for large j . So, for j k, bj N (0; =n). Thus, bj bj = n for for j k, where bj N (0; 1). Therefore, 2 b 2 = n n X b2 k i=n k+1 j n n X k i=n k+1 pn bj 2 406 22. Smoothing Using Orthogonal Functions = = n 2 X b2 k i=n k+1 j 2 2 k k since a sum of k Normals has a k distribution. Now E (k ) = k and hence E (b ) . Also, V (k ) = 2k and hence V (b ) ( =k )(2k) = (2 =k) ! 0 as n ! 1. Thus we expect b to be a consistent estimator of . There is nothing special about the choice k = n=4. Any k that increases with n at an appropriate rate will suÆce. We estimate the risk with 2 2 4 2 2 2 2 2 4 2 2 n b2 X Rb(J ) = J + b2 n j =J +1 j b2 n : + (22.27) We are now ready to give a complete description of the method, which Beran (2000) calls REACT (Risk Estimation and Adaptation by Coordinate Transformation). 22.3 Regression 407 Orthogonal Series Regression Estimator 1. Let bj 2. Let n X 1 = n Yij (xi ); j = 1; : : : ; n: i=1 n n X b2 b = k i=n k+1 j (22.28) 2 where k n=4. 3. For 1 J n, compute the risk estimate Rb n b2 X (J ) = J n + bj2 j =J +1 b2 n : + 4. Choose Jb 2 f1; : : : ng to minimize Rb(J ). 5. Let Jb X fb(x) = bj j (x): j =1 Example 22.10 Figure 22.4 shows the doppler function f and n = 2; 048 observations generated from the model Yi = r(xi ) + i N (0; (:1)2 ). Then gure shows the true where xi = i=n, i function, the data, the estimated function and the estimated risk. The estimated gure was based on Jb = 234 terms. Finally, we turn to condence bands. As before, these bands are not really for the true function r(xP) but rather for the smoothed version of the function re(x) = Jjb j j (x). =1 22. Smoothing Using Orthogonal Functions −0.4 −0.5 −0.2 0.0 0.0 0.2 0.5 0.4 408 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.04 0.00 −0.4 0.02 −0.2 0.0 Estimated Risk 0.2 0.06 0.4 0.08 0.0 0 500 1000 J FIGURE 22.4. The doppler test function. 1500 2000 22.3 Regression 409 22.11 Theorem Suppose the estimate rb is based on J terms and b is dened as in equation (22.28). Assume that J < n k + 1. An approximate 1 condence band for re is (`; u) where `(x) u(x) where = p s p s rbn(x) K J = rbn(x) + K J z b J b2 pn + n (22.29) z b J b2 pn + n (22.30) K = 1max max jj (x)j j J x and b 2 2 J b J = n 1+ k 4 (22.31) and k = n=4 is p from equation (22.28). In the cosine basis, we may use K = 2. Example 22.12 Figure 22.5 shows the condence envelope for the doppler signal. Th rst plot is based on J = 234 (the value of J that minimizes the estimated risk). The second is based on p J = 45 n. Larger J yields a higher resolution estimator at the cost of large condence bands. Smaller J yields a lower resolution estimator but has tighter condence bands. So far, we have assumed that the xi's are of the form f1=n; 2=n; : : : ; 1g. If the xi's are on interval [a; b] then we can rescale them so that are in the interval [0; 1]. If the xi 's are not equally spaced the methods we have discussed still apply so long as the xi 's \ll out" the interval [0,1] in such a way so as to not be too clumped together. If we want to treat the xi's as random instead of xed, then the method needs signicant modications which we shall not deal with here. 22. Smoothing Using Orthogonal Functions −1.0 −0.5 0.0 0.5 1.0 410 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 −1.0 −0.5 0.0 0.5 1.0 0.0 FIGURE 22.5. The condence envelope for the doppler test function using n = 2; 048 observations. The top plot shows the estimate and envelope using J = 234 terms. The bottom plot shows the estimate and envelope using J = 45 terms. 22.4 Wavelets 22.4 411 Wavelets Suppose there is a sharp jump in a regression function f at some point x but that f is otherwise very smooth. Such a function f is said to be spatially inhomogeneous. See Figure 22.6 for an example. It is hard to estimate f using the methods we have discussed so far. If we use a cosine basis and only keep low order terms, we will miss the peak; if we allow higher order terms we will nd the peak but we will make the rest of the curve very wiggly. Similar comments apply to kernel regression. If we use a large bandwidth, then we will smooth out the peak; if we use a small bandwidth, then we will nd the peak but we will make the rest of the curve very wiggly. One way to estimate inhomogeneous functions is to use a more carefully chosen basis that allows us to place a \blip" in some small region without adding wiggles elsewhere. In this section, we describe a special class of bases called wavelets, that are aimed at xing this problem. Statistical inference suing wavelets is a large and active area. We will just discuss a few of the main ideas to get a avor of this approach. We start with a particular wavelet called the Haar wavelet. The Haar father wavelet or Haar scaling function is dened by 0x<1 (x) = 10 ifotherwise (22.32) : The mother Haar wavelet is dened by (x) = 11 ifif 0 < xx 1:; (22.33) For any integers j and k dene j;k (x) = 2j= (2j x k) and j;k (x) = 2j= (2j x k): (22.34) The function j;k has the same shape as but it has been rescaled by a factor of 2j= and shifted by a factor of k. 1 2 2 1 2 2 2 22. Smoothing Using Orthogonal Functions 0.4 0.6 0.8 1.0 1.2 1.4 412 0.0 0.2 0.4 0.6 FIGURE 22.6. An inhomogeneous function. The function is smooth except for a sharp peak in one place. 0.8 1.0 22.4 Wavelets 413 See Figure 22.7 for some examples of Haar wavelets. Notice that for large j , j;k is a very localized function. This makes it possible to add a blip to a function in one place without adding wiggles elsewhere. Increasing j is like looking in a microscope at increasing degrees of resolution. In technical terms, we say that wavelets provide a multiresolution analysis of L (0; 1). Let Wj = f jk ; k = 0; 1; : : : ; 2j 1g be the set of rescaled and shifted mother wavelets at resolution j. 2 Theorem 22.13 The set of functions n ; W0 ; W1 ; W2 ; : : : ; o is an orthonormal basis for L2 (0; 1). It follows from this theorem that we can expand any function f 2 L (0; 1) in this basis. Because each Wj is itself a set of functions, we write the expansion as a double sum: 2 f (x) = (x) + where = Z 1 0 1 2X1 X j j =0 k=0 j;k f (x)(x) dx; j;k = j;k Z 1 0 (x) f (x) (22.35) j;k (x) dx: We call the scaling coeÆcient and the j;k's are called the detail coeÆcients. We call the nite sum fe(x) = (x) + j 1 J 1 2X X j =0 k=0 j;k j;k (x) (22.36) 2 0 −2 −4 −4 −2 0 2 4 22. Smoothing Using Orthogonal Functions 4 414 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 2 0 −2 −4 −4 −2 0 2 4 0.2 4 0.0 FIGURE 22.7. Some Haar wavelets. Top left: the father wavelet (x); top right: the mother wavelet (x); bottom left: 2;2 (x); bottom right: 4;10 (x). 22.4 Wavelets 415 the resolution J approximation to f . The total number of terms in this sum is 1+ J 1 X j =0 2j = 1 + 2J 1 = 2J : Example 22.14 Figure 22.8 shows the doppler signal, and its resolution J approximation for J = 3; 5 and J = 8. Haar wavelets are localized, meaning that they are zero outside an interval. But they are not smooth. This raises the question of whether there exist smooth, localized wavelets that from an orthonormal basis. In 1988, Ingrid Daubechie showed that such wavelets do exist. These smooth wavelets are diÆcult to describe. They can be constructed numerically but there is no closed form formual for the smoother wavelets. To keep things simple, we will continue to use Haar wavelets. For your interest, gure 22.9 shows an example of a smooth wavelet called the \symmlet 8." We can now use wavelets to do density estimation and regression. We shall only discuss the regression problem Yi = r(xi ) + i where i N (0; 1) and xi = i=n. To simplify the discussion we assume that n = 2J for some J . There is one major dierence between estimation using wavelets instead of a cosine (or polynomial) basis. With the cosine basis, we used all the terms 1 j J for some J . With wavelets, we use a method called thresholding where we keep a term in the function approximation if its coeÆcient is large, otherwise, we throw we don't use that term. There are many versions of thresholding. The simplest is called hard, universal thresholding. 22. Smoothing Using Orthogonal Functions 0.0 0.0 −0.4 −0.4 −0.2 −0.2 f 0.2 0.2 0.4 0.4 416 0.0 0.2 0.4 0.6 0.8 1.0 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.4 −0.2 −0.2 0.0 0.0 0.2 0.2 0.4 0.4 x 0.0 0.2 0.4 FIGURE 22.8. PThe Pdoppler signal and its reconstruction fe(x) = (x) + Jj =01 k j;k j;k (x) based on J = 3, J = 5 and J = 8. 417 −1.0 0.0 0.5 1.0 1.5 22.4 Wavelets 0 2 4 −0.2 0.2 0.6 1.0 −2 0 2 4 FIGURE 22.9. The symmlet 8 wavelet. The top plot is the mother and the bottom plot is the father. 6 418 22. Smoothing Using Orthogonal Functions Haar Wavelet Regression 1. Let J = log (n) and dene 1 X (x )Y and D = 1 X b = k i i j;k n n 2 i j;k i (xi )Yi (22.37) for 0 j J 1. 2. Estimate by p median j DJ ;k j : k = 0; : : : ; 2J b = n 0:6745 1 3. Apply universal thresholding: bj;k = 4. Set ( Dj;k 0 if jDj;kj > b otherwise: ( ) = b(x) + fb x q j 1 J 1 2X X j =j0 k=0 2 log n bj;k n ) j;k 1 1 : (22.38) (22.39) (x): In practice, we do not compute Sk and Dj;k using (22.37). Instead, we use the discrete wavelet transform (DWT) which is very fast. For Haar wavelets, the DWT works as follows. 22.4 Wavelets 419 DWT for Haar Wavelets Let y be the vector of Yi's (length n) and let J = log (n). Create a list D with elements D[[0]]; : : : ; D[[J 1]]: Set: p 2 temp < y= n: Then do: for(j m I D[[j ]] in < < < temp < (J 1) : 0)f 2j (1 : m) p temp[2 I ] temp[(2 I ) 1] = 2 p temp[2 I ] + temp[(2 I ) 1] = 2 g Remark 22.15 If you use a programming language that does not allow a 0 index for a list, then index the list as D[[1]]; : : : ; D[[J ]] and replace the second last line with p 1] = 2 The estimate for probably looks strange. It is similar to the estimate we used for the cosine basis but it is designed to be insensitive to sharp peaks in the function. To understand the intuition behind universal thresholding, consider what happens when there is no signal, that is, when j;k = 0 for all j and k. D[[j + 1]] < temp[2 I ] temp[(2 I ) 420 22. Smoothing Using Orthogonal Functions 22.16 Theorem Suppose that j;k = 0 for all j and k and let bj;k be the universal threshold estimator. Then ( P bj;k = 0 for all j; k) ! 1 as n ! 1. To simplify the proof, assume that is known. Now Dj;k N (0; =n). We will need Mill's inequality : if Z p t = N (0; 1) then P(jZ j > t) (c=t)e where c = 2= is a constant. Thus, X P(max jDj;k j > ) P(jDj;k j > ) j;k p pn X njDj;k j > P Proof. 2 2 2 = j;k X c pn exp j;k c p2 log ! 0: n 1 n 2 2 2 Example 22.17 Consider Yi = r(xi ) + i where f is the doppler signal, = :1 and n = 2048. Figure 22.10 shows the data and the estimated function using universal thresholding. Of course, the estimate is not smooth since Haar wavelets are not smooth. Nonetheless, the estimate is quite accurate. 22.5 Bibliographic Remarks A reference for orthogonal function methods is Efromovich (1999). See also Beran (2000) and Beran and Dumbgen (1998). An introduction to wavelets is given in Ogden (1997). A more advanced treatment can be found in Hardle, Kerkyacharian, Picard and Tsybakov (1998). The theory of statistical estimation 421 −0.5 0.0 0.5 22.5 Bibliographic Remarks 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 −0.4 −0.2 0.0 0.2 0.4 0.0 FIGURE 22.10. Estimate of the Doppler function using Haar wavelets and universal thresholding. 422 22. Smoothing Using Orthogonal Functions using wavelets has been developed by many authors, especially David Donoho and Ian Johnstone. There is a series of papers especially worth reading: Donoho and Johnstone (1994), Donoho and Johnstone (1995a), Donoho and Johnstone (1995b), and Donoho and Johnstone (1998). 22.6 Exercises 1. Prove Theorem 22.5. 2. Prove Theorem 22.9. 3. Let 1 1 1 1 1 1 1 2 = p3 ; p3 ; p3 ; = p2 ; p2 ; 0 ; = p6 ; p6 ; p6 : Show that these vectors have norm 1 and are orthogonal. 4. Prove Parseval's relation equation (22.6). 5. Plot the rst ve Legendre polynomials. Very, numerically, that they are orthonormal. 6. Expand the following functions in the cosine basis on [0; 1]. For (a) and (b), nd the coeÆcients j analytically. For (c) and (d), nd the coeÆcients j numerically, i.e. 1 2 j = Z 1 0 3 N r r X 1 f f (x)j (x) N N j N r=1 for some large integer N . Then plot the partial sum Pnj j j (x) for increasing values of n. p (a) f (x) = 2cos(3x). (b) f (x) = sin(x). (c) f (x) = Pj hj K (x tj ) where K (t) = (1+sign(t))=2, =1 11 =1 22.6 Exercises 423 (tj ) = (:1; :13; :15; :23; :25; :40; :44; :65; :76; :78; :81), (hj ) = (4; 5; 3; 4; 5; 4:2; 2:1; 4:3; 3:1; 2:1; 4:2). p (d) f = x(1 x) sin x : : : 7. Consider the glass fragments data. Let Y be refractive index and let X be aluminium content (the fourth variable). (a) Do a nonparametric regression to t the model Y = f (x)+ using the cosine basis method. The data are not on a regular grid. Ignore this when estimating the function. (But do sort the data rst.) Provide a function estimate, an estimate of the risk and a condence band. (b) Use the wavelet method to estimate f . 8. Show that the Haar wavelets are orthonormal. 9. Consider again the doppler signal: p 2 :1 f (x) = x(1 x) sin : x + 0:05 21 ( + 05) Let n = 1024 and let (x ; : : : ; xn ) = (1=n; : : : ; 1). Generate data Yi = f (xi ) + i where i N (0; 1). (a) Fit the curve using the cosine basis method. Plot the function estimate and condence band for J = 10; 20; : : : ; 100. (b) Use Haar wavelets to t the curve. 10. (Haar density Estimation.) Let X ; : : : ; Xn f for some density f on [0; 1]. Let's consider constructing a wavelet histogram. Let and be the Haar father and mother wavelet. Write 1 1 f (x) (x) + j 1 J 1 2X X j =0 k=0 j;k j;k (x) 424 22. Smoothing Using Orthogonal Functions where J log (n). Let 2 bj;k n X 1 = n i=1 j;k (Xi): (a) Show that bj;k is an unbiased estimate of j;k. (b) Dene the Haar histogram ( ) = (x) + fb x j 1 B 2X X j =0 k=0 bj;k j;k (x) for 0 B J 1. (c) Find an approximate expression for the MSE as a function of B . (d) Generate n = 1000 observations from a Beta (15,4) density. Estimate the density using the Haar histogram. Use leave-one-out cross vaidation to choose B . 11. In this question, we will explore the motivation for equation (22.38). Let X ; : : : ; Xn N (0; ). Let p median(jX j; : : : ; jXnj) b = n : 0:6745 (a) Show that E (b) = . (b) Simulate n = 100 observations from a N(0,1) distribution. Compute b as well as the usual estimate of . Repeat 1000 times and compare the MSE. (c) Repeat (b) but add some outliers to the data. To do this, simulate each observation from a N(0,1) with probability .95 and simulate each observation from a N(0,10) with probability .95. 12. Repeat question 6 using the Haar basis. 2 1 1 This is page 425 Printer: Opaque this 23 Classication 23.1 Introduction The problem of predicting a discrete random variable Y from another random variable X is called classication, supervised learning, discrimination or pattern recognition. In more detail, consider iid data (X1 ; Y1 ); : : : ; (Xn ; Yn) where Xi = (Xi1 ; : : : ; Xid ) 2 X R d is a d-dimensional vector and Yi takes values in some nite set Y . A classication rule is a function h : X ! Y . When we observe a new X , we predict Y to be h(X ). Example 23.1 Here is a an example with fake data. Figure 23.1 shows 100 data points. The covariate X = (X1 ; X2 ) is 2-dimensional and the outcome Y 2 Y = f0; 1g. The Y values are indicated on the plot with the triangles representing Y = 1 and the squares representing Y = 0. Also shown is a linear classication rule represented by the solid line. This is a rule of the form a + b1 x1 + b2 x2 > 0 h(x) = 10 ifotherwise : 426 23. Classication Everything above the line is classied as a 0 and everything below the line is classied as a 1. Example 23.2 The Coronary Risk-Factor Study (CORIS) data involve 462 males between the ages of 15 and 64 from three rural areas in South Africa (Rousseauw et al 1983, Hastie and Tibshirani, 1987). The outcome Y is the presence (Y = 1) or absence (Y = 0) of coronary heart disease. There are 9 covariates: systolic blood pressure, cumulative tobacco (kg), ldl (low density lipoprotein cholesterol), adiposity, famhist (family history of heart disease), typea (type-A behavior), obesity, alcohol (current alcohol consumption), and age. Figure 23.2 shows the outcomes and two of the covariates, systolic blood pressure and tobaco consumption. People with/without heart disease are coded with triangles and squares. Also shown is a linear decision boundary which we will explain shortly. In this example, the groups are much harder to tell apart. In fact, 141 people are misclassied using this classication rule. At this point, it is worth revisiting the Statistics/Data Mining dictionary: Statistics classication data covariates classier estimation Computer Science supervised learning training sample features hypothesis learning Meaning predicting a discrete Y from X (X1 ; Y1 ); : : : ; (Xn; Yn) the Xi 's map h : X ! Y nding a good classier In most cases in this Chapter, we deal with the case f0; 1g. Y = 427 0.0 0.2 0.4 x2 0.6 0.8 1.0 23.1 Introduction 0.0 0.2 0.4 0.6 x1 FIGURE 23.1. Two covariates and a linear decision boundary. 4 means Y = 1. means Y = 0. You won't see real data like this. 0.8 1.0 23. Classication 15 10 5 0 Tobacco 20 25 30 428 100 120 140 160 180 Systolic Blood Pressure FIGURE 23.2. Two covariates and and a linear decision boundary with data from the Coronary Risk-Factor Study (CORIS). 4 means Y = 1. means Y = 0. This is real life. 200 220 23.2 Error Rates and The Bayes Classier 429 23.2 Error Rates and The Bayes Classier Our goal is to nd a classication rule h that makes accurate predictions. We start with the following denitions. One can use other Denition 23.3 The true error rate of a classier h is L(h) = P(fh(X ) 6= Y g) (23.1) and the empirical error rate or training error rate is n 1X b Ln (h) = I (h(Xi ) 6= Yi ): (23.2) n i=1 First we consider the special case where Y = f0; 1g. Let r(x) = E (Y jX = x) = P(Y = 1jX = x) denote the regression function. From Bayes' theorem we have that f1 (x) r(x) = (23.3) f1 (x) + (1 )f0 (x) where f0 (x) = f (xjY = 0) f1 (x) = f (xjY = 1) and = P(Y = 1). Denition 23.4 The ned to be Bayes classication rule h is de 1 if r(x) > 12 (23.4) 0 otherwise: The set D(h) = fx : P(Y = 1jX = x) = P(Y = 0jX = x)g is called the decision boundary. h (x) = measures of error as well. 430 23. Classication Warning! The Bayes rule has nothing to do with Bayesian inference. We could estimate the Bayes rule using either frequentist or Bayesian methods. The Bayes' rule may be written in several equivalent forms: h (x) = and 1 if P(Y = 1jX = x) > P(Y = 0jX = x) 0 otherwise h (x) = 1 if f1 (x) > (1 )f0 (x) 0 otherwise: (23.5) (23.6) 23.5 The Bayes rule is optimal, that is, if h is any other classication rule then L(h ) L(h). Theorem The Bayes rule depends on unknown quantities so we need to use the data to nd some approximation to the Bayes rule. At the risk of oversimplifying, there are three main approaches: 1. Empirical Risk Minimization. Choose a set of classiers and nd bh 2 H that minimizes some estimate of L(h). H 2. Regression. Find an estimate rb of the regression function r and dene if rb(x) > 21 b h(x) = 01 otherwise : 3. Density Estimation. Estimate f0 from the Xi 's for which Yi = 0, estimate f1 from the Xi 's for which Yi = 1 and let Pn 1 b = n i=1 Yi . Dene rb(x) = Pb(Y = 1jX = x) = and b h(x) = bfb1 (x) bfb1 (x) + (1 b)fb0 (x) 1 if rb(x) > 12 0 otherwise: 23.3 Gaussian and Linear Classiers 431 Now let us generalize to the case where Y takes on more than two values as follows. Theorem 23.6 Suppose that Y 2 Y = f1; : : : ; K g. The optimal rule is h(x) = argmaxk P(Y = kjX = x) = argmaxk k fk (x) where (23.7) (23.8) fk (x)k ; (23.9) r fr (x)r r = P (Y = r), fr (x) = f (xjY = r) and argmaxk means \the value of k that maximizes that expression." P(Y = k j X = x) = P 23.3 Gaussian and Linear Classiers Perhaps the simplest approach to classication is to use the density estimation strategy and assume a parametric model for the densities. Suppose that Y = f0; 1g and that f0 (x) = f (xjY = 0) and f1 (x) = f (xjY = 1) are both multivariate Gaussians: fk (x) = 1 (2 )d=2 jk j1=2 exp 1 (x 2 k )T k 1 (x k ) ; k = 0; 1: Thus, X jY = 0 N (0 ; 0 ) and X jY = 1 N (1 ; 1 ). 432 23. Classication Theorem 23.7 If X jY = 0 N (0 ; 0 ) and X jY = 1 N (1 ; 1 ), then the Bayes rule is h (x) = ( 1 if r12 < r02 + 2 log 0 otherwise 1 0 + log jj01 jj (23.10) where ri2 = (x i )T i 1 (x i ); i = 1; 2 (23.11) is the Manalahobis distance. An equivalent way of expressing the Bayes' rule is h(x) = argmaxk Æk (x) where 1 log jk j 2 1 (x 2 k )T k 1 (x k ) + log k (23.12) and jAj denotes the determinant of a matrix A. Æk (x) = The decision boundary of the above classier is quadratic so this procedure is often called quadratic discriminant analysis (QDA). In practice, we use sample estimates of ; 1; 2 ; 0 ; 1 in place of the true value, namely: n n 1X 1X b0 = (1 Yi ); b1 = Y n i=1 n i=1 i 1 X 1 X Xi ; b1 = X b0 = n0 i: Y =0 n1 i: Y =1 i 1 X 1 X S0 = (Xi b0 )(Xi b0 )T ; S1 = (X n0 i: Y =0 n1 i: Y =1 i i i i where n0 = P i i (1 Yi ) and n1 = P i Yi . b1 )(Xi b1 )T 23.3 Gaussian and Linear Classiers 433 A simplication occurs if we assume that 0 = 0 = . In that case, the Bayes rule is h(x) = argmaxk Æk (x) (23.13) where now 1 T 1 + log k : (23.14) 2 k The parameters are estimated as before except that the mle of is n S +n S S = 0 0 1 1: n0 + n1 The classication rule is Æ1 (x) > Æ0 (x) h (x) = 10 ifotherwise (23.15) where Æk (x) = xT 1 k Æj (x) = xT S 1 bj 1 T 1 b S bj + log bj 2 j is called the discriminant function. The decision boundary fx : Æ0 (x) = Æ1(x)g is linear so this method is called linear discrimination analysis (LDA). Example 23.8 Let us return to the South African heart disease data. The decision rule in Figure 23.2 was obtained by linear discrimination. The outcome was classied as 0 classied as 1 y = 0 277 25 y = 1 116 44 The observed misclassication rate is 141=462 = :31. Including all the covariates reduces the error rate to .27. The results from quadratic discrimination are classied as 0 classied as 1 y = 0 272 30 y = 1 113 47 434 23. Classication which has about the same error rate 143=462 = :31. Including all the covariates reduces the error rate to .26. In this example, there is little advantage to QDA over LDA. Now we generalize to the case where Y takes on more than two values. 23.9 Suppose that Y 2 f1; : : : ; K g. If fk (x) = f (xjY = k) is Gaussian, the Bayes rule is Theorem h(x) = argmaxk Æk (x) where 1 log jk j 2 1 (x 2 k )T k 1 (x k ) + log k : (23.16) If the variances of the Gaussians are equal then Æk (x) = Æk (x) = xT 1 k 1 T 1 + log k : 2 k (23.17) We estimate Æk (x) by by inserting estimates of k , k and k . There is another version of linear discriminant analysis due to Fisher. The idea is to rst reduce the dimension of covariates to one dimension by projecting the data onto a line. Algebraically, this meansd replacing the covariatePX = (X1 ; : : : ; Xd ) with a linear combination U = wT X = dj=1 wj Xj . The goal is to choose the vector w = (w1 ; : : : ; wd ) that \best separates the data." Then we perform classication with the new covariate Z instead of X . We need dene what we mean by separation of the groups. We would like the two groups to have means that are far apart relative to their spread. Let j denote the mean of X for Yj and let be the variance matrix of X . Then E (U jY = j ) = J E (wT X jY = j ) = wT j and V(U ) = wT w. Dene the separa- The quantity arises in physics, where it is called the Rayleigh coeÆcient. 23.3 Gaussian and Linear Classiers tion by 435 (E (U jY = 0) E (U jY = 1))2 wT w T T (w 0 w 1 )2 = wT w T w (0 1 )(0 1 )T w = : T w wP We estimate J as follows. Let nj ni=1 I (Yi = j ) be the number of observations in group j , let X j be the sample mean vector of the X 's for group j , and let Sj be the sample covariance matrix in group j . Dene wT S w (23.18) Jb(w) = T B w SW w where SB = (X 0 X 1 )(X 0 X 1 ) (n0 1)S0 + (n1 1)S1 : SW = (n0 1) + (n1 1) Theorem 23.10 The vector w = SW1 (X 0 X 1 ) (23.19) is a minimizer of Jb(w). We call (23.20) U = wT X = (X 0 X 1 )T SW1 X the Fisher linear discriminant function. The midpoint m between X 0 and X 1 is 1 1 m = (X 0 + X 1 ) = (X 0 X 1 )T SB 1 (X 0 + X 1 ) (23.21) 2 2 Fisher's classication rule is 0 if wT X m h(x) = 1 if wT X < m 0 if (X 0 X 1 )T SW1 x m = 1 if (X 0 X 1 )T SW1 x < m: Fisher's rule is the same as the Bayes linear classier in equation (23.14) when b = 1=2. J (w) = 436 23. Classication 23.4 Linear Regression and Logistic Regression A more direct approach to classication is to estimate the regression function r(x) = E (Y jX = x) without bothering to estimste the densities fk . For the rest of this section, we will only consider the case where Y = f0; 1g. Thus, r(x) = P(Y = 1jX = x) and once we have an estimate rb, we will use the classication rule if rb(x) > 21 h(x) = 01 otherwise (23.22) : The simplest regression model is the linear regression model Y = r(x) + = 0 + d X j =1 j Xj + (23.23) where E () = 0. This model can't be correct since it does not force Y = 0 or 1. Nonetheless, it can sometimes lead to a decent classier. Recall that the least squares estimate of = (0 ; 1 ; : : : ; d )T minimizes the residual sums of squares RSS ( ) = n X i=1 Yi 0 d X j =1 2 Xij j : Let us briey review what that estimator is. Let X denote the N (d + 1) matrix of the form 2 6 X=6 6 4 1 1 .. . 1 X11 X21 .. . Xn1 ::: ::: .. . ::: X1d X2d .. . Xnd 3 7 7 7: 5 Also let Y = (Y1 ; : : : ; Yn)T . Then, RSS ( ) = (Y X )T (Y X ) 23.4 Linear Regression and Logistic Regression 437 and the model can be written as Y = X + where = (1 ; : : : ; n )T . The least squares solution b that minimizes RSS can be found by elementary calculus and is given by b = (XT X) 1 XT Y: The predicted values are b b = X: Y P Now we use (23.22) to classify, where rb(x) = b0 + j bj xj . It seems sensible to use a regression method that takes into account that Y 2 f0; 1g. The most common method for doing so is logistic regression. Recall that the model is P e0 + P x : r(x) = P (Y = 1jX = x) = 1 + e 0 + x j j j j j j (23.24) We may write this is logit P (Y = 1jX = x) = 0 + X j j xj where logit (a) = log(a=(1 a)). Under this model, each Yi is a Bernoulli with success probability P e0 + P X pi ( ) = : 1 + e0 + X j j ij j j ij The likelihood function for the data set is L( ) = n Y i=1 pi ( )Y (1 pi ( ))1 We obtain the mle numerically. i Yi : 438 23. Classication Example 23.11 Let us return to the heart disease data. A logistic regression yields the following estimates and Wald tests for the coeÆcients: Covariate Estimate Std. Error Wald statistic p-value Intercept -6.145 1.300 -4.738 0.000 sbp 0.007 0.006 1.138 0.255 tobacco 0.079 0.027 2.991 0.003 ldl 0.174 0.059 2.925 0.003 adiposity 0.019 0.029 0.637 0.524 famhist 0.925 0.227 4.078 0.000 typea 0.040 0.012 3.233 0.001 obesity -0.063 0.044 -1.427 0.153 alcohol 0.000 0.004 0.027 0.979 age 0.045 0.012 3.754 0.000 Are surprised by the fact that systolic blood pressure is not signicant or by the minus sign for the obesity coeÆcient? If yes, then you are confusing correlation and causation. There are lots of unobserved confounding variables so the coeÆcients above cannot be interpreted causally. The fact that blood pressure is not signicant does not mean that blood pressure is not an important cause of heart disease. It means that it is not an important predictor of heart disease relative to the other variables in the model. The error rate, using this model for classication, is .27. The error rate from a linear regression is .26. We can get a better classier by tting a richer model. For example we could t X X logit P (Y = 1jX = x) = 0 + j xj + jk xj xk : (23.25) j j;k More generally, we could add terms of up to order r for some integer r. Large values of r give a more complicated model which should t the data better. But there is a bias-variance tradeo which we'll discuss later. Example 23.12 If we use model (23.25) for the heart disease data with r = 2, the error rate is reduced to .22. 23.5 Relationship Between Logistic Regression and LDA 439 The logistic regression model can easily be extended to k groups but we shall not give the details here. 23.5 Relationship Between Logistic Regression and LDA LDA and logistic regression are almost the same thing. If we assume that each group is Gaussian with the same covariance matrix then we saw that P(Y = 1jX = x) 1 = log 0 ( + 1 )T 1 (1 0 ) + xT 1 (1 log P(Y = 0jX = x) 1 2 0 0 + T x: On the other hand, the logistic model is, by assumption, P(Y = 1jX = x) = 0 + T x: log P(Y = 0jX = x) These are the same model since they both lead to classication rules that are linear in x. The dierence is in how we estimate the parameters. The joint density of a single observation is f (x; y ) = f (xjy )f (y ) = f (y jx)f (x). In LDA we estimated the whole joint distribution by estimating f (xjy ) and f (y ); specically, we estimated fk (x) = f (xjY = k), 1 = fY (1) and 0 = fY (0). We maximized the likelihood Y i f (xi ; yi) = Y | i f (xi jyi) {z Y i f (yi ) : (23.26) } | {z } Gaussian Bernoulli In logistic regression we maximized the conditional likelihood Q i f (yi jxi ) but we ignored the second term f (xi ): Y i f (xi ; yi) = Y | i f (yi jxi ) {z logistic Y i f (xi ) : } | {z ignored } (23.27) 0 ) 440 23. Classication Since classication only requires knowing f (y jx), we don't really need to estimate the whole joint distribution. Here is an analogy: we can estimate the mean of a distribution with the sample mean X without bothering to estimate the whole density function. Logistic regression leaves the marginal distribution f (x) un-specied so it is more nonparametric than LDA. To summarize: LDA and logistic regression both lead to a linear classication rule. In LDA we estimate the entire joint distribution f (x; y ) = f (xjy )f (y ). In logistic regression we only estimate f (y jx) and we don't bother estimating f (x). 23.6 Density Estimation and Naive Bayes Recall that the Bayes rule is h(x) = argmaxk k fk (x): If we can estimate k and fk then we can estimate the Bayes classication rule. Estimating k is easy but what about fk ? We did this previously by assuming fk was Gaussian. Another strategy is to estimate fk with some nonparametric density estimator fbk such as a kernel estimator. But if x = (x1 ; : : : ; xd ) is high dimensional, nonparametric density estimation is not very reliable. This problem is ameliorated if we assume that X1 ; : : : ; Xd Qd are independent, for then, fk (x1 ; : : : ; xd ) = j =1 fkj (xj ). This reduces the problem to d one-dimensional density estimation problems, within each of the k groups. The resulting classier is called the naive Bayes classier. The assumption that the components of X are independent is usually wrong yet the resulting classier might still be accurate. Here is a summary of the steps in the naive Bayes classier: 23.7 Trees 441 The Naive Bayes Classier 1. For each group k, compute an estimate fbkj of the density fkj for Xj , using the data for which Yi = k. 2. Let fbk (x) 3. Let = fbk (x1 ; : : : ; xd ) = d Y j =1 fbkj (xj ): n 1X bk = I (Y = k) n i=1 i where I (Yi = k) = 1 if Yi = k and I (Yi = k) = 0 if Yi 6= k. 4. Let h(x) = argmaxk bk fbk (x): The naive Bayes classier is especially popular when x is high dimensional and discrete. In that case, fbkj (xj ) is especially simple. 23.7 Trees Trees are classication methods that partition the covariate space X into disjoint pieces and then classify the observations according to which partition element they fall in. As the name implies, the classier can be represented as a tree. For illustration, suppose there are two covariates, X1 = age and X2 = blood pressure. Figure 23.3 shows a classication tree using these variables. The tree is used in the following way. If a subject has Age 50 then we classify him as Y = 1. If a subject has Age < 50 then we check his blood pressure. If blood pressure is < 100 442 23. Classication Age < 50 50 1 Blood Pressure < 100 100 0 1 FIGURE 23.3. A simple classication tree. then we classify him as Y = 1, otherwise we classify him as Y = 0. Figure 23.4 shows the same classier as a partition of the covariate space. Here is how a tree is constructed. For simplicity, we focus on the case in which 2 Y = f0; 1g. First, suppose there is only a single covariate X . We choose a split point t that divides the real line into two sets A1 = ( 1; t] and A2 = (t; 1). Let pbs (j ) be the proportion of observations in As such that Yi = j : pbs (j ) = Pn I (Yi = j; Xi As ) i=1 Pn i=1 I (Xi As ) 2 2 (23.28) for s = 1; 2 and j = 0; 1. The impurity of the split t is dened to be I (t) = where s = 1 2 X s=1 1 X j =0 s pbs (j )2 : (23.29) (23.30) 23.7 Trees Blood Pressure 1 110 1 0 50 Age FIGURE 23.4. Partition representation of classication tree. 443 444 23. Classication This measure of impurity is known as the Gini index. If a partition element As contains all 0's or all 1's, then s = 0. Otherwise, s > 0. We choose the split point t to minimize the impurity. (Other indices of impurity besides can be used besides the Gini index.) When there are several covariates, we choose whichever covariate and split that leads to the lowest impurity. This process is continued until some stopping criterion is met. For example, we might stop when every partition element has fewer than n0 data points, where n0 is some xed number. The bottom nodes of the tree are called the leaves. Each leaf is assigned a 0 or 1 depending on whether there are more data points with Y = 0 or Y = 1 in that partition element. This procedure is easily generalized to the case where Y 2 f1; : : : ; K g: We simply dene the impurity by s = 1 k X j =1 pbs (j )2 (23.31) where pbi (j ) is the proportion of observations in the partition element for which Y = j . Example 23.13 Figure 23.5 shows a classication tree for the heart disease data. The misclassication rate is .21. Suppose we do the same example but only using two covariates: tobacco and age. The misclassication rate is then .29. The tree is shown in Figure 23.6. Because there are only two covariates, we can also display the tree as a partition of X = R 2 as shown in Figure 23.7. Our description of how to build trees is incomplete. If we keep splitting until there are few cases in each leaf of the tree, we are likely to overt the data. We should choose the complexity of the tree in such a way that the estimated true error rate is low. In the next section, we discuss estimation of the error rate. 23.7 Trees 445 age < 31.5 | age < 50.5 tobacco < 0.51 alcohol < 11.105 typea < 68.5 famhist:a 0 ldl < 6.705 tobacco < 7.605 0 0 0 1 typea < 42.5 adiposity < 28.955 tobacco < 4.15 1 adiposity < 24.435 0 adiposity < 28 1 1 1 typea < 48 0 0 0 0 FIGURE 23.5. A classication tree for the heart disease data. 1 446 23. Classication age < 31.5 | age < 50.5 tobacco < 0.51 tobacco < 7.47 0 0 0 0 FIGURE 23.6. A classication tree for the heart disease data using two covariates. 1 447 1 0 5 10 15 0 0 0 tobacco 20 25 30 23.7 Trees 0 20 30 40 age FIGURE 23.7. The tree pictured as a partition of the covariate space. 50 60 448 23. Classication 23.8 Assessing Error Rates and Choosing a Good Classier How do we choose a good classier? We would like to have a classier h with a low true error rate L(h). Usually, we can't use the training error rate Lbn (h) as an estimate of the true error rate because it is biased downward. Example 23.14 Consider the heart disease data again. Suppose we t a sequence logistic regression models. In the rst model we include one covariate. In the second model we include two covariates and so on. The ninth model includes all the covariates. We can go even further. Let's also t a tenth model that includes all nine covariates plus the rst covariate squared. Then we t an eleventh model that includes all nine covariates plus the rst covariate squared and the second covariate squared. Continuing this way we will get a sequence of 18 classiers of increasing complexity. The solid line in Figure 23.8 shows the observed classication error which steadily decreases as we make the model more complex. If we keep going, we can make a model with zero observed classication error. The dotted line shows the cross-validation estimate of the error rate (to be explained shortly) which is a better estimate of the true error rate than the observed classication error. The estimated error decreases for a while then increases. This is the bias-variance tradeo phenomenon we have seen before. There are many ways to estimate the error rate. We'll consider two: cross-validation and probability inequalities. Cross-Validation. The basic idea of cross-validation, which we have already encountered in curve estmation, is to leave out some of the data when tting a model. The simplest version of cross-validation involves randomply splitting the data into two 449 0.30 0.28 0.26 error rate 0.32 0.34 23.8 Assessing Error Rates and Choosing a Good Classier 5 10 Number of terms in the model FIGURE 23.8. Solid line is the observed error rate and dashed line is the cross-validation estimate of true error rate. 15 450 23. Classication Training Data | {z b h T Validation Data } | {z Lb FIGURE 23.9. Cross-validation. pieces: the training set T and the validation set V . Often, about 10 per cent of the data might be set aside as the validation set. The classier h is constructed from the training set. We then estimate the error by Lb(h) = 1 X I (h(Xi ) 6= YI ): m X 2V (23.32) i where m is the size of the validation set. See Figure 23.9. Another approach to cross-validation is K-fold cross-validation which is obtained from the following algorithm. V } 23.8 Assessing Error Rates and Choosing a Good Classier 451 K -fold cross-validation. 1. Randomly divide the data into K chunks of approximately equal size. A common choice is K = 10. 2. For k = 1 to K do the following: (a) Delete chunk k from the data. (b) Compute the classier bh(k) from the rest of the data. (c) Use bh(k) to the predict the data in chunk k. Let Lb(k) denote the observed error rate. 3. Let Lb(h) K 1X = Lb : K k=1 (k) (23.33) This is how dotted line in Figure 23.8 was computed. An alternative approach to K-fold cross-validation is to split the data into two pieces called the training set and the test set. All the models are t on the training set. The models are then used to do predictions on the test set and the errors are estimated from these predictions. When the clasication procedure is time consuming and there are lots of data, this approach is preferred to K-fold cross-validation. Cross-validation can be applied to any classication method. To apply it to trees, one begins by tting an initial tree. Smaller trees are obtained by pruning tree, that is removing splits from the tree. We can do this for trees of various sizes where size refers to the number of terminal nodes on the tree. Cross-validation is then used to estimate error rate as a function of tree size. Example 23.15 Figure 23.10 shows the cross-validation plot for a classication tree tted to the heart disease data. The error 452 23. Classication is shown as a function of tree size. Figure 23.11 shows the tree with the size chosen by minimizing the cross-validation error. . Another approach to estimating the error rate is to nd a condence interval for Lbn (h) using probability inequalities. This method is useful in the context of empirical risk minimization. Let H be a set of classiers, for example, all linear classiers. Empirical risk minimization means choosing the classier bh 2 H to minimize the training error Lbn (h), also called the empirical risk. Thus, Probability Inequalities ! 1X b h = argminh2H Lbn (h) = argminh2H I (h(Xi ) 6= Yi ) : n i (23.34) b b b Typically, Ln (h) underestimates the true error rate L(h) because b h was chosen to make Lbn (bh) small. Our goal is to assess how much underestimation is taking place. Our main tool for this analysis is Hoeding's inequality. Recall that if X1 ; : : : ; Xn Bernoulli(p), then, for any > 0, j P ( pb 2 pj > ) 2e 2n (23.35) P where pb = n 1 ni=1 Xi . First, suppose that H = fh1 ; : : : ; hm g consists of nitely many classiers. For any xed h, Lbn (h) converges in almost surely to L(h) by the law of large numbers. We will now establish a stronger result. 23.16 (Uniform Convergence.) Assume H is nite and has m elements. Then, Theorem P max j h2H Lbn (h) L(h)j > 2me 2n2 : We will use Hoeding's inequality and weSwill also use the fact that if A1 ; : : : ; Am is a set of events then P( m i=1 Ai ) Proof. 453 145 140 135 130 score 150 155 160 23.8 Assessing Error Rates and Choosing a Good Classier 2 4 6 8 size FIGURE 23.10. Cross validation plot 10 12 14 454 23. Classication age < 31.5 | age < 50.5 0 typea < 68.5 famhist:a tobacco < 7.605 0 1 1 0 FIGURE 23.11. Smaller classication tree with size chosen by cross-validation. 1 23.8 Assessing Error Rates and Choosing a Good Classier Pm i=1 P(Ai ). Now, b n (h) L(h) P max L h2H j j> = Theorem 23.17 Let s = Then Lbn (bh) is a 1 Proof. P (j P [ 455 ! jLbn(h) L(h)j > h2H X b P Ln (h) L(h) > H 2H X 2 2 2e 2n = 2me 2n : j j H 2H 2 2m log : n condence interval for L(bh). This follows from the fact that j > ) P max j h2H 2me 2n = : Lbn (bh) L(bh) Lbn (bh) j> L(bh) 2 When H is large the condence interval for L(bh) is large. The more functions there are in H the more likely it is we have \overt" which we compensate for by having a larger condence interval. In practice we usually use sets H that are innite, such as the set of linear classiers. To extend our analysis to these cases we want to be able to say something like P sup j h2H Lbn (h) L(h)j > something not too big: All the other results followed from this inequality. One way to develop such a generalization is by way of the VapnikChervonenkis or VC dimension. We now consider the main ideas in VC theory. 456 23. Classication Let A be a class of sets. Give a nite set F = fx1 ; : : : ; xn g let n NA (F ) = # F \ o A: A2A (23.36) be the number of subsets of F \picked out" by A. Here #(B ) denotes the number of elements of a set B . The shatter coefcient is dened by s(A; n) = max NA (F ) (23.37) F 2Fn where Fn consists of all nite sets of size n. Now let X1 ; : : : ; Xn P and let 1X I (Xi 2 A) Pn (A) = n i denote the empirical probability measure. The following remarkable theorem bounds the distance between P and Pn . 23.18 (Vapnik and Chervonenkis (1971).) For any P, n and > 0, Theorem P sup jPn (A) A2A j > 8s(A; n)e P(A) n2 =32 : (23.38) The proof, though very elegant, is long and we omit it. If H is a set of classiers, dene A to be the class of sets of the form fx : h(x) = 1g. When then dene s(H; n) = s(A; n). Theorem 23.19 P A1 sup jLbn (h) h2H L(h)j > 8s(H; n)e n2 =32 : condence interval for L(bh) is Lbn (bh) n where 32 8s(H; n) 2n = log n : 23.8 Assessing Error Rates and Choosing a Good Classier 457 These theorems are only useful if the shatter coeÆcients do not grow too quickly with n. This is where VC dimension enters. Denition 23.20 The VC (Vapnik-Chervonenkis) dimen- sion of a class of sets A is dened as follows. If s(A; n) = 2n for all n set V C (A) = 1. Otherwise, dene V C (A) to be the largest k for which s(A; n) = 2k . Thus, the VC-dimension is the size of the largest nite set F than can be shattered by A meaning that A picks out each subset of F . If H is a set of classiers we dene V C (H) = V C (A) where A is the class of sets of the form fx : h(x) = 1g as h varies in H. The following theorem shows that if A has nite VC-dimension then the shatter coeÆcients grow as a polynomial in n. Theorem 23.21 If A has nite VC-dimension v , then s(A; n) nv + 1: A = f( 1; a]; a 2 Rg. The A shatters ever one point set fxg but it shatters no set of the form fx; y g. Therefore, V C (A) = 1. Example 23.23 Let A be the set of closed intervals on the real line. Then A shatters S = fx; y g but it cannot shatter sets with 3 points. Consider S = fx; y; zT g where x < y < z. One cannot nd an interval A such that A S = fx; z g. So, V C (A) = 2. Example 23.24 Let A be all linear half-spaces on the plane. Any Example 23.22 Let three point set (not all on a line) can be shattered. No 4 point set can be shattered. Consider, for example, 4 points forming a diamond. Let T be the left and rightmost points. This can't be picked out. Other congurations can also be seen to be unshatterable. So V C (A) = 3. In general, halfspaces in Rd have VC dimension d + 1. 458 23. Classication Example 23.25 Let A be all rectangles on the plane with sides parallel to the axes. Any 4 point set can be shattered. Let S be a 5 point set. There is one point that is not leftmost, rightmost, uppermost, or lowermost. Let T be all points in S except this point. Then T can't be picked out. So V C (A) = 4. 23.26 Let x have dimension d and let H be th set of linear classiers. The VC dimension of H is d +1. Hence, a 1 condence interval for the true error rate is Lb(bh) where Theorem 8(nd+1 + 1) 32 2n = log n : 23.9 Support Vector Machines In this section we consider a class of linear classiers called support vector machines. Throughout this section, we assume that Y is binary. It will be convenient to label the outcomes as 1 and +1 instead of 0 and 1. A linear classier can then be written as h(x) = sign H (x) where x = (x1 ; : : : ; xd ), H (x) = a0 + and sign(z ) = 8 < : d X i=1 ai xi 1 if z < 0 0 if z = 0 1 if z > 0: First, suppose that the data are linearly separable, that is, there exists a hyperplane that perfectly separates the two classes. 23.9 Support Vector Machines 459 Lemma 23.27 The data can be separated by some hyperplane if P and only if there exists a hyperplane H (x) = a0 + di=1 ai xi such that Yi H (xi ) 1; i = 1; : : : ; n: (23.39) Suppose the data can be separated by a hyperplane Pd W (x) = b0 + i=1 bi xi . It follows that there exists some constant c such that Yi = 1 implies W (Xi ) c and Yi = 1 implies W (XP i ) c. Therefore, Yi W (Xi ) c for all i. Let H (x) = a0 + di=1 ai xi where aj = bj =c. Then Yi H (Xi ) 1 for all i. The reverse direction is straightforward. Proof. In the separable case, there will be many separating hyperplanes. How should we choose one? Intuitively, it seems reasonable to choose the hyperplane \furthest" from the data in the sense that it separates the +1'a and -1's and maximizes the distance to the closest point. This hyperplane is called the maximum margin hyperplane. The margin is the distance to from the hyperplane to the nearest point. Points on the boundary of the margin are called support vectors. See Figure 23.12. 23.28 The hyperplane Hb (x) = ba0 + Pd ai xi i=1 b that separatesP the data and maximizes the margin is given by minimizing (1=2) dj=1 b2j subject to (23.39). Theorem It turns out that this problem can be recast as a quadratic programming problem. Recall that hXi ; Xk i = XiT Xk is the inner product of Xi and Xk . 23.29 Let Hb (x) = ba0 + Pd ai xi i=1 b denote the optimal (largest margin) hyperplane. Then, for j = 1; : : : ; d, Theorem b aj = n X i=1 bi Yi Xj (i) 460 23. Classication c b c b bc c b bc bc bc bc c b c b c b c b bc H (x) = a0 + aT x = 0 bc FIGURE 23.12. The hyperplane H (x) has the largest margin of all hyperplanes that separate the two classes. where Xj (i) is the value of the covariate Xj for the ith data point, and b = (b1 ; : : : ; bn) is the vector that maximizes n X i=1 i n X n 1X Y Y hX ; X i 2 i=1 k=1 i k i k i k subject to (23.40) i 0 and 0= X i i Yi : The points Xi for which b 6= 0 are called support vectors. ba0 can be found by solving bi Yi(XiT ba + b0 = 0 for any support point Xi . Hb may be written as Hb (x) = ba0 + n X i=1 bi Yi hx; Xi i: 23.10 Kernelization 461 There are many software packages that will solve this problem quickly. If there is no perfect linear classier, then one allows overlap between the groups by replacing the condition (23.39) with Yi H (xi ) 1 i ; i 0; i = 1; : : : ; n: (23.41) The variables 1 ; : : : ; n are called slack variables. We now maximize (23.40) subject to 0 i c; i = 1; : : : ; n and n X i=1 i Yi = 0: The constant c is a tuning parameter that controls the amount of overlap. 23.10 Kernelization There is a trick for improving a computationally simple classier h called kernelization. The idea is to map the covariate X { which takes values in X { into a higher dimensional space Z and apply the classier in the bigger space Z . This can yield a more exible classier while retaining computationally simplicity. The standard example of this idea is illustrated in Figure 23.13. The covariate x = (x1 ; x2 ). The Yi 's can be separated into two groups using an ellipse. Dene a mapping by p z = (z1 ; z2 ; z3 ) = (x) = (x21 ; 2x1 x2 ; x22 ): This maps X = R 2 into Z = R 3 . In the higher dimensional space Z , the Yi 's are seperable by a linear decision boundary. In other words, a linear classier in a higher dimenional space corresponds to a non-linear classier in the original space. 462 23. Classication x2 + ++ z2 + + + + + + x1 + + + + + + + + + + + + + z1 + z3 FIGURE 23.13. Kernelization. Mapping the covariates into a higher dimensional space can make a complicated decision boundary into a simpler decision boundary. 23.10 Kernelization 463 There is a potential drawback. If we signicantly expand the dimension of the problem, we might increase the computational burden. For example, x has dimension d = 256 and we wanted to use all fourth order terms, then z = (x) has dimension 183,181,376. What saves us is two observations. First, many classiers do not require that we know the values of the individual points but, rather, just the inner product between pairs of points. Second, notice in our example that the inner product in Z can be written hz; zei = h(x); (xe)i = x21 xe21 + 2x1 xe1 x2 xe2 + x22 xe22 = (hx; xei)2 k(x; xe): Thus, we can compute hz; zei without ever computing Zi = (Xi ). To summarize, kernelization means involves nding a mapping : X ! Z and a classier such that: 1. Z has higher dimension than X of classiers. and so leads a richer set 2. The classier only requires computing inner products. 3. There is a function K , called a kernel, such that h(x); (xe)i = K (x; xe). 4. Everywhere the term hx; xei appears in the algorithm, replace it with K (x; xe). In fact, we never need to construct the mapping at all. We only need to specify a kernel k(x; xe) that corresponds to h(x); (xe)i for some . This raises an interesting question: given a function of two variables K (x; y ), does there exist a function (x) such that K (x; y ) = h(x); (y )i. The answer is provided 464 23. Classication by Mercer's theorem which says, roughly, that if K is positive denite { meaning that Z Z K (x; y )f (x)f (y )dxdy 0 for square integrable functions f { then such a exists. Examples of commonly used kernels are: r polynomial K (x; xe) = hx; xei + a sigmoid K (x; xe) = tanh( ahx; xei + b) 2 2 Gaussian K (x; xe) = exp jjx xejj =(2 ) Let us now see how we can use this trick in LDA and in support vector machines. Recall that the Fisher linear discriminant method replaces X with U = wT X where w is chosen to maximize the Rayleigh coeÆcient wT S w J (w) = T B ; w SW w SB = (X 0 X 1 )(X 0 X 1 )T and (n0 1)S0 (n1 1)S1 SW = + : (n0 1) + (n1 1) (n0 1) + (n1 1) In the kernelized version, we replace Xi with Zi = (Xi ) and we nd w to maximize J (w) = where and SeB = (Z 0 wT SeB w wT SeW w Z 1 )(Z 0 ! (23.42) Z 1 )T ! (n0 1)Se0 (n1 1)Se1 SW = + : (n0 1) + (n1 1) (n0 1) + (n1 1) 23.10 Kernelization 465 Here, Sej is the sample of covariance of the Zi 's for which Y = j . However, to take advantage of kernelization, we need to reexpress this in terms of inner products and then replace the inner products with kernels. It can be shown that the maximizing vector w is a linear combination of the Zi 's. Hence we can write w= Also, n X i=1 i Zi : n 1 X Zj = (Xi)I (Yi = j ): nj i=1 Therefore, wT Z j = n X i Zi T i=1 n X n 1 X n 1 X (Xi )I (Yi = j ) nj i=1 I (Y = j )ZiT (Xs) nj i=1 s=1 i s n n X 1 X I (Y = j )(Xi)T (Xs) = nj i=1 i s=1 s n n X 1 X = i I (Ys = j )K (Xi ; Xs) nj i=1 s=1 = T Mj = where Mj is a vector whose ith component is Mj (i) = It follows that where M = (M0 can write n 1 X K (Xi ; Xs)I (Yi = j ): nj s=1 wT SeB w = T M M1 )(M0 M1 )T . By similar calcuations, we wT SeW w = T N 466 23. Classication where 1 1 N = K0 I 1 K0T + K1 I 1 K1T ; n0 n1 I is the identity matrix, 1 is a matrix of all one's, and Kj is the n nj matrix with entries (Kj )rs = K (xr ; xs ) with xs varying over the observations in group j . Hence, we now nd to maximize T M J ( ) = T : N Notice that all the qauntities are expressed in terms of the kernel. Formally, the solution is = N 1 (M0 M1 ). However, N might be non-invertible. In this case on replaces N by N + bI form some constant b. Finally, the projection onto the new subspace can be written as U= wT (x) = n X i=1 i K (xi ; x): The support vector machine can similarly be kernelized. We simply replace hXi ; Xj i with K (Xi ; Xj ). For example, instead of maximizing (23.40), we now maximize n X i=1 i n X n 1X Y Y K (Xi ; Xj ): 2 i=1 k=1 i k i k (23.43) Pn bi Yi K (X; Xi ). i=1 The hyperplane can be written as Hb (x) = ba0 + 23.11 Other Classiers There are many other classiers and space precludes a full dicussion of all of them. Let us briey mention a few. The k-nearest-neighbors classier is very simple. Given a point x, nd the k data points closest to x. Classify x using the majority vote of these k neighbors. Ties can be broke randomly. The parameter k can be chosen by cross-validation. 23.11 Other Classiers 467 Bagging is a method for reducing the variability of a clas- sier. It is most helpful for highly non-linear classiers such as a tree. We draw B boostrap samples from the data. The bth bootstrap sample yields a classier hb . The nal classier is b h(x) = P 1 if B1 Bb=1 hb (x) 12 0 otherwise: Boosting is a method for starting with a simple classier and gradually improving it by retting the data giving higher weight to misclassied samples. Suppose that H is a collection of classiers, for example, tress with onely one split. Assume that Yi 2 f 1; 1g and that each h is such that h(x) 2 f 1; 1g. We usually give equal weight to all data points in the methods we have discussed. But one can incorporate unequal weights quite easily in most algorithms. For example, in constructing a tree, we could replace the impurity measure with a weighted impurity measure. The original version of bossting, called AdaBoost, is as follows. 1. Set the weights wi = 1=n, i = 1; : : : ; n. 2. For j = 1; : : : ; J , do the following steps: (a) Constructing a classier hj from the data using the weights w1 ; : : : ; wn. (b) Compute the weighted error estimate: Lbj = Pn i I (Yi = hj (Xi )) i=1 wP : n i=1 wi 6 (c) Let j = log((1 Lbj )=Lbj ). (d) Update the weights: wi wi e I (Y 6=h (X )) j i j i 468 23. Classication 3. The nal classier is b h(x) = sign J X j =1 j hj (x) : There is now an enormous literature trying to explain and improve on boosting. Whereas bagging is a variance reduction technique, boosting can be thought of as a bias reduction technique. We starting with a simple { and hence highly biased { classier, and we gradually reduce the bias. The disadvantage of boosting is that the nal classier is quite complicated. 23.12 Bibliographic Remarks An excellent reference on classication is Hastie, Tibshirani and Friedman (2001). For more on the theory, see Devroye, Gyor, Lugosi (1996) and Vapnik (2001). 23.13 Exercises 1. Prove Theorem 23.5. 2. Prove Theorem 23.7. 3. Download the spam data from: http://www-stat.stanford.edu/tibs/ElemStatLearn/index.html The data le can also be found on the course web page. The data contain 57 covariates relating to email messages. Each email message was classied as spam (Y=1) or not spam (Y=0). The outcome Y is the last column in the le. The goal is to predict whether an email is spam or not. 23.13 Exercises 469 (a) Construct classication rules using (i) LDA, (ii) QDA, (iii) logistic regression and (iv) a classication tree. For each, report the observed misclassication error rate and construct a 2-by-2 table of the form Y =0 Y =1 b h(x) = 0 bh(x) = 1 Hint: Here are some things to remember when using R. (i) To t a logistic regression and get the classied values: d <- data.frame(x,y) n <- nrow(x) out <- glm(y ~ .,family=binomial,data=d) p <- out$fitted.values pred <- rep(0,n) pred[p > .5] <- 1 table(y,pred) (ii) To build a tree you must turn y into a factor and you must do it before you make the data frame: y <- as.factor(y) d <- data.frame(x,y) library(tree) out <- tree(y ~ .,data=d) print(summary(out)) plot(out,type="u",lwd=3) text(out) pred <- predict(out,d,type="class") table(y,pred) 470 23. Classication (b) Use 5-fold cross-validation to estimate the prediction accuracy of LDA and logistic regression. (c) Sometimes it helps to reduce the number of covariates. One strategy is to compare Xi for the spam and email group.. For each of the 57 covariates, test whether the mean of the covriate is the same or dierent between the two groups. Keep the 10 covariates with the smallest pvalues. Try LDA and logstic regression using only these 10 variables. 4. Let A be the set of two-dimensional spheres. That is, A 2 A if A = f(x; y) : (x a)2 + (y b)2 c2g for some a; b; c. Find the VC dimension of A. 5. Classify the spam data using support vector machines. Free software for the support vector machine is at http://svmlight.joachims.org/ It is very easy to use. After installing, do the following. First, you must recode the Yi's as +1 and -1. Then type svm_learn data output where \data" is the le with the data. It expects the data in the following form: 1 1:3.4 2:.17 3:129 etc where the rst number is Y (+1 or -1) and the rest of the line means: X1 is 3.4, X2 is .17, etc. If the le \test" contains test data, then type 23.13 Exercises 471 svm_classify test output predictions to get the predictions on the test data. There are many options available, as explained on the web site. 6. Use VC theory to get a condence interval on the true error rate of the LDA classier in question 3. Suppose that the covariate X = (X1 ; : : : ; Xd ) has d dimensions. Assume that each variable has a continuous distribution. 7. Suppose that Xi 2 R and that Yi = 1 whenever jXi j 1 and Yi = 0 whenever jXi j > 1. Show that no linear classier can perfectly classify these data. Show that the kernelized data Zi = (Xi ; Xi2 ) can be linearly separated. 8. Repeat question 5 using the kernel K (x; xe) = (1 + xT xe)p . Choose p by cross-validation. 9. Apply the k nearest neighbors classier to the \iris data." Get the data in R from the command data(iris). Choose k by cross-validation. 10. (Curse of Dimensionality.) Suppose that X has a uniform distribution on the d-dimensional cube [ 1=2; 1=2]d . Let R be the distance from the origin to the closest neighbor. Show that the median of R is 0 @ where 1 1=d 1 1=n 2 A 1 vd (1) d=2 vd (r) = ((d=2) + 1) is the volume of a sphere of radius r. When does R exceed the edge of the cube when n = 100; 1000; 10000? rd 472 23. Classication 11. Fit a tree to the data in question 3. Now apply bagging and report your results. 12. Fit a tree that uses only one split on one variable to the data in question 3. Now apply boosting. 13. Let r(x) = P(Y = 1jX = x) and let rb(x) be an estimate of r(x). Consider the classier h(x) = 1 if rb(x) 1=2 0 otherwise: Assume that rb(x) N (r (x); 2 (x)) for some functions r(x) and 2 (x)). Show that 0 P(Y 6= h(X )) 1 B @ sign (rb(x) 1 (1=2))(r(x) (x) (1=2)) where is the standard Normal cdf. Regard sign (rb(x) (1=2))(r(x) (1=2)) as a type of bias term. Explain the implications for the bias-variance trade-o in classication (Friedman, 1997). C A This is page 473 Printer: Opaque this 24 Stochastic Processes 24.1 Introduction Most of this book has focused on iid sequences of random variables. Now we consider sequences of dependent random variables. For example, daily temperatures will form a sequence of time-ordered random variables and clearly the temperature on one day is not independent of the temperature on the previous day. A stochastic process fXt : t 2 T g is a collection of random variables. We shall sometimes write X (t) instead of Xt . The variables Xt take values in some set X called the state space. The set T is called the index set and for our purposes can be thought of as time. The index set can be discrete T = f0; 1; 2; : : :g or continuous T = [0; 1) depending on the application. Example 24.1 (iid observations.) A sequence of iid random vari- ables can be written as fXt : t 2 T g where T = f1; 2; 3; : : : ; g. 474 24. Stochastic Processes Thus, a sequence of iid random variables is an example of a stochastic process. Example 24.2 (The Weather.) Let X = fsunny; cloudyg. A typi- cal sequence (depending on where you live) might be sunny; sunny; cloudy; sunny; cloudy; cloudy; This process has a discrete state space and a discrete index set. Example 24.3 (Stock Prices.) Figure 24.1 shows the price of a stock over time. The price is monitored continuously so the index set T is not discrete. Price is discrete but, for practical purposes we can imagine treating it as a continuous variable. Example 24.4 (Empirical Distribution Function.) Let X1 ; : : : ; Xn F where F is some cdf on [0,1]. Let 1X I (Xi t) Fbn (t) = n i=1 n be the empirical cdf . For any xed value t, Fbn (t) is a random variable. But the whole empirical cdf n o Fbn (t) : t 2 [0; 1] is a stochastic process with a continuous state space and a continuous index set. We end this section by recalling a basic fact. If X1 ; : : : ; Xn are random variables then we can write the joint density as f (x1 ; : : : ; xn ) = f (x1 )f (x2 jx1 ) f (xn jx1 ; : : : ; xn 1 ) n Y = f (xi jpasti ) (24.1) i=1 where pasti refers to all the variables before Xi . 7 8 9 price 10 11 12 13 24.1 Introduction 0 2 4 6 8 10 time FIGURE 24.1. Stock price over ten week period. 475 476 24. Stochastic Processes 24.2 Markov Chains The simplest stochastic process is a Markov chain in which the distribution of Xt depends only on Xt 1 . In this section we assume that the state space is discrete, either X = f1; : : : ; N g or X = f1; 2; : : : ; g and that the index set is T = f0; 1; 2; : : :g. Typically, most authors write Xn instead of Xt when discussing Markov chains and I will do so as well. Denition 24.5 The process fXn : n chain if P(Xn 2 T g is a Markov = x j X0 ; : : : ; Xn 1 ) = P(Xn = x j Xn 1 ) (24.2) for all n and for all x 2 X . For a Markov chain, equation (24.1) simplifes to f (x1 ; : : : ; xn ) = f (x1 )f (x2 jx1 )f (x3 jx2 ) f (xn jxn 1 ): A Markov chain can be represented by the following DAG: X1 X2 X3 Xn Each variable has a single parent, namely, the previous observation. The theory of Markov chains is a very rich and complex. We have to get through many denitions before we can do anything interesting. Our goal is to answer the following questions: 1. When does a Markov chain \settle down" into some sort of equilibrium? 2. How do we estimate the parameters of a Markov chain? 24.2 Markov Chains 477 3. How can we construct Markov chains that converge to a given equilibrium distribution and why would we want to do that? We will answer questions 1 and 2 in this chapter. We will answer question 3 in the next chapter. To understand question 1, look at the two chains in Figure 24.2. The rst chain oscillates all over the place and will continue to do so forever. The second chain eventually settles into an equilibrium. If we constructed a histogram of the rst process, it would keep changing as we got more and more observations. But a histogram from the second chain would eventually converge to some xed distribution. The key quantities of a Markov chain are the probabilities of jumping from one state into another state. Transition Probabilities. Denition 24.6 We call P(Xn+1 = j jXn = i) (24.3) the transition probabilities. If the transition probabilities do not change with time, we say the chain is homogeneous. In this case we dene pij = P(Xn+1 = j jXn = i). The matrix P whose (i; j ) element is pij is called the transition matrix. We will only consider homogeneous P chains. Notice that P has two properties: (i) pij 0 and (ii) i pij = 1. Each row is a probability mass function. A matrix with these properties is called a stochastic matrix. Example 24.7 (Random Walk With Absorbing Barriers.) Let f1; : : : ; N g. X = Suppose you are standing at one of these points. 24. Stochastic Processes −50 −150 −100 x 0 478 0 200 400 600 800 1000 600 800 1000 4 2 0 −2 x 6 8 10 time 0 200 400 time FIGURE 24.2. The rst chain does not settle down into an equilibrium. The second does. 24.2 Markov Chains 479 Flip a coin with P(Heads) = p and P(T ails) = q = 1 p. If it is heads, take one step to the right. If it is tails, take one step to the left. If you hit one of the endpoints, stay there. The transition matrix is 2 6 6 6 P=6 6 6 4 3 0 0 q p 0 0 77 0 q 0 p 0 0 7 7 77 : 1 0 0 0 0 0 0 0 0 0 0 0 0 0 q 0 p 5 0 0 Example 24.8 Suppose the state space is 1 X = fsunny; cloudyg. Then X1 ; X2 ; : : : represents the weather for a sequence of days. The weather today clearly depends on yesterday's weather. It might also depend on the weather two days ago but as a rst approximation we might assume that the dependence is only one day back. In that case the weather is a Markov chain and a typical transition matrix might be Sunny Cloudy Sunny 0.4 0.6 Cloudy 0.8 0.2 For example, if it is sunny today, there is a 60 per cent chance it will be cloudy tommorrow. Let pij (n) = P(Xm+n = j jXm = i) (24.4) be the probability of of going from state i to state j in n steps. Let Pn be the matrix whose (i; j ) element is pij (n). These are called the n-step transition probabilities. Theorem 24.9 (The Chapman-Kolmogorov equations.) n-step probabilities satisfy pij (m + n) = X k pik (m)pkj (n): The (24.5) 480 24. Stochastic Processes Proof. Recall that, in general, P(X = x; Y = y ) = P(X = x)P(Y = y jX = x): This fact is true in the more general form P(X = x; Y = y jZ = z ) = P(X = xjZ = z )P(Y = y jX = x; Z = z ): Also, recall the law of total probability: X P(X = x) = P(X = x; Y = y ): y Using these facts and the Markov property we have pij (m + n) = P(Xm+n = j jX0 = i) X = P(Xm+n = j; Xm = k jX0 = i) = = = k X k X k X k P(Xm+n = j jXm = k; X0 = i)P(Xm = kjX0 = i) P(Xm+n = j jXm = k)P(Xm = kjX0 = i) pik (m)pkj (n): Look closely at equation (24.5). This is nothing more than the equation for matrix multiplication. Hence we have shown that Pm+n = Pm Pn : (24.6) By denition, P1 = P. Using the above theorem, P2 = P1+1 = 2 P1 P1 = PP = P . Continuing this way, we see that Pn = Pn | P P {z P} multiply the matrix n times : (24.7) 24.2 Markov Chains 481 Let n = (n(1); : : : ; n(N )) be a row vector where n (i) = P(Xn = i) (24.8) is the marginal probability that the chain is in state i at time n. In particular, 0 is called the initial distribution. To simulate a Markov chain, all you need to know is 0 and P. The simulation would look like this: Draw X0 0 . Thus, P(X0 = i) = 0 (i). Step 2: Suppose the outcome of step 1 is i. Draw X1 P. In other words, P(X1 = j jX0 = i) = pij . Step 3: Suppose the outcome of step 2 is j . Draw X2 P. In other words, P(X2 = kjX1 = j ) = pjk . And so on. Step 1: It might be diÆcult to understand the meaning of n. Imaging simulating the chain many times. Collect all the outcomes at time n from all the chains. This histogram would look approximately like n . A consequence of theorem 24.9 is the following. Lemma 24.10 The marginal probabilities are given by n = 0 Pn : Proof. n (j ) = P(Xn = j ) X = P(Xn = j jX0 = i)P (X0 = i) i X = 0 (i)pij (n) = 0 Pn : i 482 24. Stochastic Processes Summary 1. Transition matrix: P(i; j ) = P(Xn+1 = j jXn = i). 2. n-step matrix: Pn(i; j ) = P(Xn+m = j jXm = i). 3. Pn = Pn . 4. Marginal: n (i) = P(Xn = i). 5. n = 0 Pn . The states of a Markov chain can be classied according to various properties. States. Denition 24.11 We say that i j (or j is accessible from i) if pij (n) > 0 for some n, and we write i ! j . If i ! j and j ! i then we write i $ j and we say that i and j communicate. reaches The communication relation satises the following properties: Theorem 24.12 1. i $ i. 2. If i $ j then j $ i. 3. If i $ j and j 4. $ k then i $ k. The set of states X can be written as a disjoint union S S of classes X = X X where two states i and j 1 2 communicate with each other if and only if they are in the same class. 24.2 Markov Chains 483 If all states communicate with each other then the chain is called irreducible. A set of states is closed if, once you enter that set of states you never leave. A closed set consisting of a single state is called an absorbing state. Example 24.13 Let X = f1; 2; 3; 4g and 0 B B P=B B @ 1 0 0 C 0 0C C 1 1 C 4 4 A 0 0 0 1 1 3 2 3 1 4 2 3 1 3 1 4 The classes are f1; 2g; f3g and f4g. State 4 is an absorbing state. Suppose we start a chain in state i. Will the chain ever return to state i? If so, that state is called persistent or recurrent. Denition 24.14 State i is P(Xn 24.15 or persistent if = i for some n 1 j X0 = i) = 1: Otherwise, state i is Theorem recurrent transient . A state i is recurrent if and only if X n pii (n) = 1: (24.9) A state i is transient if and only if X n Proof. Dene In = pii (n) < 1: 1 if Xn = i 0 if Xn 6= i: (24.10) 484 24. Stochastic Processes P The number of times that the chain is in state i is Y = 1 n=0 In . The mean of Y , given that the chain starts in state i, is 1 X E (Y jX0 = i) = E (In jX0 = i) n=0 = = 1 X n=0 1 X n=0 = ijX0 = i) P(Xn pii (n): Dene ai = P(Xn = i for some n 1 j X0 = i). If i is recurrent, ai = 1. Thus, the chain will eventually return to i. Once it does return to i, we argue again that since ai = 1, the chain will return to state i again. By repeating this argument, we conclude that E (Y jX0 = i) = 1. If i is transient, then ai < 1. When the chain is in state i, there is a probability 1 ai > 0 that it will never return to state i. Thus, the probability that the chain is in state i exactly n times is ani 1 (1 ai ). This is a geometric distribution which has nite mean. Theorem 24.16 Facts about recurrence. 1. If state i is recurrent and i $ j then j is recurrent. 2. If state i is transient and i $ j then j is transient. 3. A nite Markov chain must have at least one recurrent state. 4. The states of a nite, irreducible Markov chain are all recurrent. Theorem X 24.17 (Decomposition Theorem.) can be written as the disjoint union X = XT [ X [ 1 X 2 The state space 24.2 Markov Chains where XT are the transient states and each ducible set of recurrent states. 485 Xi is a closed, irre- To discuss the convregence of chains, we need a few more denitions. Suppose that X0 = i. Dene the recurrence time Convergence of Markov Chains. Tij = minfn > 0 : Xn = j g (24.11) assuming Xn ever returns to state i, otherwise dene Tij = 1. The mean recurrence time of a recurrent state i is X mi = E (Tii ) = nfii (n) (24.12) n where fij (n) = P (X1 6= j; X2 6= j; : : : ; Xn 1 6= j; Xn = j jX 0 = i): A recurrent state is null if mi = 1 otherwise it is called nonnull or positive. Lemma 24.18 If a state is null and recurrent, then pnii ! 0. Lemma 24.19 In a nite state Markov chain, all recurrent states are positive. Consider a three state chain with transition matrix 2 3 0 1 0 4 0 0 1 5: 1 0 0 Suppose we start the chain in state 1. Then we will be in state 3 at times 3, 6, 9, . . . . This is an example of a periodic chain. Formally, the period of state i is d if pii (n) = 0 whenever n is not divisible by d and d is the largest integer with this property. Thus, d = gcdfn : pii (n) > 0g where gcd means \greater common divisor." State i is periodic if d(i) > 1 and aperiodic if d(i) = 1. A state with period 1 is called aperiodic. 486 24. Stochastic Processes Lemma 24.20 If state i has period d and i $ j then j has period d. Denition 24.21 A state is ergodic if it is recurrent, non- null and aperiodic. A chain is ergodic if all its states are ergodic. Let = (i : i 2 X ) be a vector of non-negative numbers that sum to one. Thus can be thought of as a probability mass function. Denition 24.22 We say that is a variant ) distribution if = P. stationary (or in- Here is the intuition. Draw X0 from distribution and suppose that is a stationary distribution. Now draw X1 according to the transition probability of the chain. The distribition of X1 is then 1 = 0 P = P = . The distribution of X2 is P2 = ( P)P = P = . Continuing this way, we see that the distribution of Xn is Pn = . In other words: If at any time the chain has distribution then it will continue to have distribution forever. 24.2 Markov Chains Denition 24.23 We say that a chain has tribution if 2 6 6 n P ! 6 4 .. . 487 limiting dis- 3 7 7 7 5 for some . In other words, j = limn!1 Pnij exists and is independent of i. Here is the main theorem. An irreducible, ergodic Markov chain has a unique stationary distribution . The limiting distrubution exists and is equal to . If g is any bounded function, then, with probability 1, Theorem lim N !1 24.24 N 1X N n=1 g (Xn ) ! E (g ) X j g (j )j : (24.13) The last statement of the theorem, is the law of large numbers for Markov chains. It says that sample averages converge to their expectations. Finally, there is a special condition which will be useful later. We say that satises detailed balance if i pij = pjij : (24.14) Detailed balance guarantees that is a stationary distribution. If satises detailed balance then is a stationary distribution. Theorem 24.25 th P We need P to show that P P = . The j element of P is i i pij = i j pji = j i pji = j : Proof. 488 24. Stochastic Processes The importance of detailed balance will become clear when we discuss Markov chain Monte Carlo methods. Just because a chain has a stationary distribution does not mean it converges. Warning! Example 24.26 Let 2 3 0 1 0 P = 4 0 0 1 5: 1 0 0 Let = (1=3; 1=3; 1=3). Then P = so is a stationary distribution. If the chain is started with the distribution it will stay in that distribution. Imagine simulating many chains and checking the marginal distribution at each time n. It will always be the uniform distribution . But this chain does not have a limit. It continues to cycle around forever. Examples of Markov Chains Example 24.27 (Random Walk) . Let X = f: : : ; 2; 1; 0; 1; 2; : : : ; g and suppose that pi;i+1 = p, pi;i 1 = q = 1 p. All states communicate hence either all the states are recurrent or all are transient. To see which, suppose we start at X0 = 0. Note that 2n n n p00 (2n) = p q n (24.15) since, the only way to get back to 0 is to have n heads (steps to the right) and n tails (steps to the left). We can approximate this expression using Stirling's formula which says that p n! nn ne n p 2: 24.2 Markov Chains 489 Inserting this approximation into (24.15) shows that p00 (2n) P (4pq )n pn : P It is easy to check that n p00 (n) < 1 if and only if n p00 (2n) < 1. Moreover, Pn p00 (2n) = 1 if and only if p = q = 1=2. By Theorem (24.15), the chain is recurrent if p = 1=2 otherwise it is transient. Example 24.28 Let X = f1; 2; 3; 4; 5; 6g. Let 2 1 1 0 0 6 21 32 6 4 4 0 0 6 1 1 1 1 6 6 4 4 4 4 P= 6 6 41 0 41 14 6 60 0 0 0 4 0 0 0 0 3 0 0 7 0 07 7 0 07 7 7 1 0 4 7 7 1 7 1 2 2 5 1 2 1 2 Then C1 = f1; 2g and C2 = f5; 6g are irreducible closed sets. States 3 and 4 are transient because the of the path 3 ! 4 ! 6 and once you hit state 6 you cannot return to 3 or 4. Since pii (1) > 0, all the states are aperiodic. In summary, 3 and 4 are transient while 1,2,5 and 6 are ergodic. Example 24.29 (Hardy-Weinberg.) Here is a famous example from genetics. Suppose a gene can be type A or type a. There are three types of people (called genotypes): AA, Aa and aa. Let (p; q; r) denote the fraction of people of each genotype. We assume that everyone contributes one of their two copies of the gene at random to their children. We also assume that mates are selected at random. The latter is not realistic however, it is often reasonable to assume that you do not choose your mate based on whether they are AA, Aa or aa. (This would be false if the gene was for eye color and if people chose mates based on eye color.) Imagine if we pooled eveyone's genes together. The proportion 490 24. Stochastic Processes of A genes is P = p + (q=2) and the proportion of a genes is Q = r + (q=2). A child is AA with probability P 2, aA with probability 2P Q and aa with probability Q2 . Thus, the fraction of A genes in this generation is q 2 q q P2 + PQ = p + + p+ r+ : 2 2 2 However, r = 1 p q . Substitute this in the above equation and you get P 2 + P Q = P . A similar calculation shows that fraction of a genes is Q. We have shown that the proportion of type A and type a is P and Q and this remains stable after the rst generation. The proportion of people of type AA, Aa, aa is thus (P 2 ; 2P Q; Q2) from the second generation and on. This is called the Hardy-Weinberg law. Assume everyone has exactly one child. Now consider a xed person and let Xn be the genotype of their nth descendent. This is a Markov chain with state space X = fAA; Aa; aag. Some basic calculations will show you that the transition matrix is 2 4 P Q P 2 P +Q 0 P 2 0 Q 2 Q 3 5: The stationary distribution is = (P 2; 2P Q; Q2 ). Consider a chain with nite state space X = f1; 2; : : : ; N g. Suppose we observe n observations X1 ; : : : ; Xn from this chain. The unknown parameters of a Markov chain are the initial probabilities 0 = (0 (1); 0(2); : : : ; ) and the elements of the transition matrix P. Each row of P is a multinomial distribution. So we are essentially estimating N distributions (plus the initial probabilities). Let nij be the observed number of transitions from state i to state j . The likelihood function is n N Y N Y Y L(0; P) = 0(x0 ) pX 1;X = 0(x0 ) pnij : Inference for Markov Chains. ij r=1 r r i=1 j =1 24.3 Poisson Processes 491 There is only one observation on 0 so we can't estimate that. Rather, we focus on estimating P. The mle is obtained by maximizing L(0 ; P) subject to the constraint that the elements are non-negative and the rows sum to 1. The solution is pbij = nij ni P where ni = Nj=1 nij . Here we are assuming that ni > 0. If note, then we set pbij = 0 by convention. Theorem 24.30 (Consistency and Asymptotic Normality of the mle .) Assume that the chain is ergodic. Let pbij (n) denote the mle after n observations. Then pbij (n) P! pij . Also, hp Ni (n)(pbij i pij ) N (0; ) P where the left hand side is a matrix, Ni (n) = nr=1 I (Xr = i) and 8 < pij (1 pij ) (i; j ) = (k; `) ij;k` = p p i = k; j 6= ` : 0 ij i` otherwise: 24.3 Poisson Processes One of the most studied and useful stochastic processes is the Poisson process. It arises when we count occurences of events over time, for example, traÆc accidents, radioactive decay, arrival of email messages etc. As the name suggests, the Poisson process is intimately related to the Poisson distribution. Let's rst review the Poisson distribution. Recall that X has a Poisson distribution with parameter { written X Poisson() { if P (X = x) p(x; ) = e x ; x = 0; 1; 2; : : : x! 492 24. Stochastic Processes Also recall that E (X ) = and V (X ) = . If X Poisson(), Y Poisson( ) and X q Y , then X + Y Poisson( + ). Finally, if N Poisson() and Y jN = n Binomial(n; p), then the marginal distribution of Y is Y Poisson(p). Now we describe the Poisson process. Imaging you are at your computer. Each time a new email message arrives you record the time. Let Xt be the number of messages you have received up to and including time t. Then, fXt : t 2 [0; 1)g is a stochastic process with state space X = f0; 1; 2; : : :g. A process of this form is called a counting process. A Poisson process is a counting process that satises certain conditions. In what follows, we will sometimes write X (t) instead of Xt . Also, we need the following notation. Write f (h) = o(h) if f (h)=h ! 0 as h ! 0. This means that f (h) is smaller than h when h is close to 0. For example, h2 = o(h). 24.3 Poisson Processes Denition 24.31 A 493 is a stochastic process fXt : t 2 [0; 1)g with state space X = f0; 1; 2; : : :g such that Poisson process 1. X (0) = 0. 2. For any 0 = t0 < t1 < t2 < < tn , the increments X (t1 ) X (t0 ); X (t2 ) X (t1 ); ; X (tn) X (tn 1 ) are independent. 3. There is a function (t) such that P(X (t + h) P(X (t + h) We call (t) the X (t) = 1) = (t)h + o(h) (24.16) X (t) 2) = o(h): (24.17) intensity function . The last condition means that the probability of an event in [t; t + h] is approximately h(t) while the probability of more than one event is small. Theorem 24.32 tion (t), then If Xt is a Poisson process with intensity func- X (s + t) X (s) Poisson(m(s + t) where m(t) = Z t 0 m(s)) (s) ds: In particular, X (t) Poisson(m(t)). Hence, E (X (t)) = m(t) and V (X (t)) = m(t). 494 24. Stochastic Processes Denition 24.33 A Poisson process with intensity func- tion (t) for some > 0 is called a homogeneous Poisson process with rate . In this case, X (t) Poisson(t): Let X (t) be a homogeneous Poisson process with rate . Let Wn be the time at which the nth event occurs and set W0 = 0. The random variables W0 ; W1 ; : : : ; are called waiting times. Let Sn = Wn+1 Wn . Then S0 ; S1 ; : : : ; are called sojourn times or interarraival times. The sojourn times S0 ; S1 ; : : : are iid random variables. Their distribution is exponential with mean 1=, that is, they have density Theorem 24.34 f (s) = e s ; s 0: The waiting time Wn Gamma(n; 1=) i.e. it has density f (w) = 1 n n 1 w e (n) t : Hence, E (Wn ) = n= and V (Wn ) = n=2 . Proof. First, we have P(S1 > t) = P(X (t) = 0) = e t with shows that the cdf for S1 is 1 e for S1 . Now, j P(S2 > t S1 t . This shows the result = s) = P(no events in (s; s + t]jS1 = s) = P(no events in (s; s + t]) (increments are independent) = e t : 24.3 Poisson Processes 495 Hence, S2 has an exponential distribution and is independent of S1 . The result follows by repeating the argument. The result for Wn follows since a sum of exponentials has a Gamma distribution. Example 24.35 Figures 24.3 and 24.4 show requests to a WWW server in Calgary.1 Assuming that this is a homogeneous Poisson process, N X (T ) Poisson(T ). The likelihood is L() / e T (T )N which is maxmimized at N b = = 48:0077 T in units per minute. In the preceding example we might want to check if the assumption that the data follow a Poisson process is reasonable. To proceed, let us rst note that if a Poisson process is observed on [0; T ] then, given N = X (T ), the event times W1 ; : : : ; WN are a sample from a Uniform(0; T ) distribution. To see intutively why this is true, note that the probability of nding an event in a small interval [t; t + h] is proportional to h. But this doesn't depend on t. Since the probability is constant over t it must be uniform. Figure 24.5 shows a histogram and a some density estimates. The density does not appear to be uniform. To test this formally, let is divide into 4 equal length intervals I1 ; I2 ; I3 ; I4 . Let pi be the probability of a point being in Ii . The null hypothesis is that p1 = p2 = p3 = p4 . We can test this hypothesis using either a likelihood ratio test or a 2 test. The latter is 4 X (Oi Ei )2 i=1 Ei 1 See http://ita.ee.lbl.gov/html/contrib/Calgary-HTTP.html for more information. 24. Stochastic Processes 0.6 0.8 1.0 1.2 1.4 496 0 2 4 6 0.6 0.8 1.0 1.2 1.4 time 0 200 400 600 800 time FIGURE 24.3. Hits on a web server. First plot is rst 20 events. Second plot is several hours worth of data. 1000 1200 497 10 5 X(t) 15 20 24.3 Poisson Processes 24/09 24/09 24/09 24/09 24/09 24/09 600 200 0 X(t) 1000 time (seconds) 24/09 24/09 25/09 time (seconds) FIGURE 24.4. X (t) for hits on a web server. First plot is rst 20 events. Second plot is several hours worth of data. 25/09 25/09 498 24. Stochastic Processes where Oi is the number of observations in Ii and Ei = n=4 is the expected number under the null. This yields 2 = 252 which a p-value of 0. Strong evidence againt the null. 24.4 Hidden Markov Models 24.5 Bibliographic Remarks This is standard material and there are many good references. In this Chapter, I followed the following references quite closely: Probability and Random Processes by Grimmett and Stirzaker (1982), An Introduction to Stochastic Modeling by Taylor and Karlin (1998), Stochastic Modeling of Scientic Data by Guttorp (1995) and Probability Models for Computer Science by Ross (2002). Some of the exercises are from these texts. 24.6 Exercises 1. Let X0 ; X1 ; : : : be a Markov chain with states f0; 1; 2g and transition matrix 2 3 0:1 0:2 0:7 P = 4 0:9 0:1 0:0 5 0:1 0:8 0:1 Assume that 0 = (0:3; 0:4; 0:3). Find P(X0 = 0; X1 = 1; X2 = 2) and P(X0 = 0; X1 = 1; X2 = 1). 2. Let Y1 ; Y2; : : : be a sequence of iid observations such that P(Y = 0) = 0:1, P(Y = 1) = 0:3, P(Y = 2) = 0:2, P(Y = 3) = 0:4. Let X0 = 0 and let Xn = maxfY1; : : : ; Yng: Show that X0 ; X1 ; : : : is a Markov chain and nd the transition matrix. 24.6 Exercises density(x = w/60) 0.0015 0.0005 0.0010 Density 150 100 0 0.0000 50 Frequency 200 Histogram of w/60 0 200 400 600 800 1000 1200 0 w/60 0.000 0.001 0.002 0.003 0.004 density(x = w/60, bw = "ucv") Density 499 0 200 400 600 800 1000 1200 FIGURE 24.5. Density estimate of event times. 500 1000 1500 500 24. Stochastic Processes 3. Consider a two state Markov chain with states X = f1; 2g and transition matrix 1 a a P= b 1 b where 0 < a < 1 and 0 < b < 1. Prove that b a n a +b a +b lim P = b : a n!1 a+b a+b 4. Consider the chain from question 3 and set a = :1 and b = :3. Simulate the chain. Let n 1X pbn (1) = I (Xi = 1) n i=1 n 1X I (Xi = 2) pbn (2) = n i=1 be the proportion of times the chain is in state 1 and state 2. Plot pbn (1) and pbn (2) versus n and verify that they converge to the values predicted from the answer in the previous question. 5. An important Markov chain is the branching process which is used in biology, genetics, nuclear physics and many other elds. Suppose that an animal has Y children. P1 Let pk = P (Y = k). Hence, pk 0 for all k and k=0 pk = 1. Assume each animal has the same lifespan and that they produce ospring according to the distribution pk . Let Xn be the number of animals in the nth generation. Let Y1(n) ; : : : ; YX(n) be the ospring produced in the nth generation. Note that n Xn+1 = Y1(n) + + YX(n) : n Let = E (Y ) and = V (Y ). Assume throughout this question that X0 = 1. Let M (n) = E (Xn ) and V (n) = V (Xn ). 2 24.6 Exercises 501 (a) Show that M (n+1) = M (n) and V (n+1) = 2 M (n)+ 2 V (n). (b) Show that M (n) = n and that V (n) = 2 n 1 (1 + + + n 1 ). (c) What happens to the variance if > 1? What happens to the variance if = 1? What happens to the variance if < 1? (d) The population goes extinct if Xn = 0 for some n. Let us thus dene the extinction time N by N = minfn : Xn = 0g: Let F (n) = P (N N . Show that F (n) = n) be the cdf of the random variable 1 X k=0 pk (F (n 1))k ; n = 1; 2; : : : Hint: Note that the event fN ng is the same as event fXn = 0g. Thus, P(fN ng) = P(fXn = 0g). Let k be the number of ospring of the original parent. The population becomes extinct at time n if and only if each of the k sub-populations generated from the k ospring goes extinct in n 1 generations. (e) Suppose that p0 = 1=4, p1 = 1=2, p2 = 1=4. Use the formula from (5d) to compute the cdf F (n). 502 24. Stochastic Processes 6. Let 2 3 0:40 0:50 0:10 P = 4 0:05 0:70 0:25 5 0:05 0:50 0:45 Find the stationary distribution . 7. Show that if i is a recurrent state and i $ j , then j is a recurrent state. 8. Let 2 1 3 0 13 0 0 13 3 6 21 14 14 0 0 0 7 7 6 60 0 0 0 1 0 7 6 7 P= 6 41 14 14 0 0 14 7 7 6 40 0 1 0 0 0 5 0 0 0 0 0 1 Which states are transient? Which states are recurrent? 9. Let 0 1 P = 1 0 Show that = (1=2; 1=2) is a stationary distribution. Does this chain converge? Why/why not? 10. Let 0 < p < 1 and q = 1 2 q 6 6q P =6 q 6 4q 1 p. Let 3 p 0 0 0 0 p 0 07 7 0 0 p 07 7 0 0 0 p5 0 0 0 0 Find the limiting distribution of the chain. 11. Let X (t) be an inhomogeneous PoissonR process with intensity function (t) > 0. Let (t) = 0t (u)du. Dene Y (s) = X (t) where s = (t). Show that Y (s) is a homogeneous Poisson process with intensity = 1. 24.6 Exercises 503 12. Let X (t) be a Poisson process with intensity . Find the conditional distribution of X (t) given that X (t + s) = n. 13. Let X (t) be a Poisson process with intensity . Find the probability that X (t) is odd, i.e. P(X (t) = 1; 3; 5; : : :). 14. Suppose that people logging in to the University computer system is described by a Poisson process X (t) with intensity . Assume that a person stays logged in for some random time with cdf G. Assume these times are all independent. Let Y (t) be the number of people on the system at time t. Find the distribution of Y (t). 15. Let X (t) be a Poisson process with intensity . Let W1 ; W2 ; : : : ; be the waiting times. Let f be an arbitrary function. Show that 0 1 Zt X (t) X [email protected] f (Wi )A = f (w)dw: i=1 0 16. A two dimensional Poisson point process is a process of random points on the plane such that (i) for any set A, the number of points falling in A is Poisson with mean (A) where (A) is the area of A, (ii) the number of events in nonoverlapping regions is independent. Consider an arbitrary point x0 in the plane. Let X denote the distance from x0 to the nearest random point. Show that 2 P(X > t) = e t and E (X ) = p1 2 : This is page 505 Printer: Opaque this 25 Simulation Methods In this we will see that by generating data in a clever way, we can solve a number of problems such as integrating or maximizing a complicated function.1 For integration, we will study three methods: (i) basic Monte Carlo integration, (ii) importance sampling and (iii) Markov chain Monte Carlo (MCMC). 25.1 Bayesian Inference Revisited Simulation methods are especially useful in Bayesian inference so let us briey review the main ideas in Bayesian inference. Given a prior f () and data X n = (X1 ; : : : ; Xn ) the posterior density is f (jX n) = R 1 My main source for this chapter is and G. Casella. L()f () L(u)f (u)du Monte Carlo Statistical Methods by C. Robert 506 25. Simulation Methods where L() is the likelihood function. The posterior mean is = Z f (j X n)d R = R L()f ()d L()f ()d : If = (1 ; : : : ; k ) is multidimensional, then we might be interested in the posterior for one of the components, 1 , say. This marginal posterior density is f (1 j X n) = Z Z Z f ( ; : : : ; k jX n)d dk 1 2 which involves high dimensional integration. You can see that integrals play a big role in Bayesian inference. When is high dimensional, it may not be feasible to calculate these integrals analytically. Simulation methods will often be very helpful. 25.2 Basic Monte Carlo Integration R Suppose you want to evaluate the integral I = ab h(x)dx for some function h. If h is an \easy" function like a polynomial or triginometric function then we can do the integral in closed form. In practice, h can be very complicated and there may be no known closed form expression for I . There are many numerical techniques for evaluating I such as Simpson's rule, the trapezoidal rule, Guassian quadrature and so on. In some cases these techniques work very well. But other times they may not work so well. In particular, it is hard to extend them to higher dimensions. Monte Carlo integration is another approach to evaluating I which is notable for its simplicity, generality and scalability. Let us begin by writing I= Z b a h(x)dx = Z b a h(x)(b a) 1 b a dx = Z b a w(x)f (x)dx 25.2 Basic Monte Carlo Integration 507 where w(x) = h(x)(b a) and f (x) = 1=(b a). Notice that f is the density for a uniform random variable over (a; b). Hence, I = Ef (w(X )) where X Unif(a; b). Suppose we generate X1 ; : : : ; XN large. By the law of large numbers Ib 1 N N X i=1 Unif(a; b) where N is p w(Xi ) ! E (w(X )) = I: This is the basic Monte Carlo integration method. We can also compute the standard error of the estimate s se b = p N where P Ib)2 N 1 where Yi = w(Xi). A 1 condence interval for I is Ib z=2 se b. We can take N as large as we want and hence make the length of the condence inteval very small. s = 2 i (Yi Let's try this on an example where we know the R1 3 3 true answer. Let h(x) = x . Hence, I = 0 x dx = 1=4. Based on N = 10; 000 I got Ib = :2481101 with a standard error of .0028. Not bad at all. Example 25.1 A simple generalization of the basic method is to consider integrals of the form I= Z h(x)f (x)dx where f (x) is a probability density function. Taking f to be a Unif (a,b) gives us the special case above. The only dierence is 508 25. Simulation Methods that now we draw X1 ; : : : ; XN Ib 1 N f and take N X i=1 h(Xi ) as before. Example 25.2 Let 1 2 f (x) = p e x =2 2 be the standard Normal pdf. Suppose we want to compute the cdf at some point x: I= Z x 1 f (s)ds = (x): Of course, we could look this up in a table or use the pnorm command in R. But suppose you don't have access to either. Write Z I = h(s)f (s)ds where 1 s<x 0 s x: Now we generate X1 ; : : : ; XN N (0; 1) and set 1X number of observations Ib = h(Xi ) = N i N h(s) = x: For example, with x = 2, the true answer is (2) = :9772 and the Monte Carlo estimate with N = 10; 000 yields .9751. Using N = 100; 000 I get .9771. Let X Binomial(n; p1 ) and Y Binomial(m; p2 ). We would like to estimate Æ = p2 p1 . The mle is Æb = pb2 pb1 = (Y=m) (X=n). We can get the standard error se b using the delta method (remember?) which yields Example 25.3 (Two Binomials) r se b = pb1 (1 pb1 ) pb2 (1 pb2 ) + n m 25.2 Basic Monte Carlo Integration 509 and then construct a 95 per cent condence interval Æb 2 se b. Now consider a Bayesian analysis. Suppose we use the prior f (p1 ; p2 ) = f (p1 )f (p2 ) = 1, that is, a at prior on (p1 ; p2 ). The posterior is f (p1 ; p2 jX; Y ) / pX1 (1 p1 )n X pY (1 2 p2 )m Y : The posterior mean of Æ is Æ= Z 1 0 Z 0 1 Æ (p1 ; p2 )f (p1 ; p2 jX; Y ) = Z 1 Z 0 0 1 (p2 p1 )f (p1 ; p2 jX; Y ): If we want the posterior density of Æ we can rst get the posterior cdf Z F (cjX; Y ) = P (Æ cjX; Y ) = f (p1 ; p2 jX; Y ) A where A = f(p1 ; p2 ) : p2 p1 cg. The density can then be obtained by dierentiating F . To avoid all these integrals, let's use simulation. Note that f (p1 ; p2 jX; Y ) = f (p1 jX )f (p2 jY ) which implies that p1 and p2 are independent under the posterior distribution. Also, we see that p1 jX Beta(X +1; n X +1) and p2 jY Beta(Y +1; m Y + 1). Hence, we can simulate (P1(1) ; P2(1) ); : : : ; (P1(N ) ; P2(N ) ) from the posterior by drawing P1(i) P2(i) Beta(X + 1; n X + 1) Beta(Y + 1; m Y + 1) for i = 1; : : : ; N . Now let Æ (i) = P2(i) Æ 1 N X i P1(i) . Then, Æ (i) : We can also get a 95 per cent posterior interval for Æ by sorting the simulated values, and nding the :025 and :975 quantile. The posterior density f (Æ jX; Y ) can be obtained by applying density 510 25. Simulation Methods estimation techniques to Æ (1) ; : : : ; Æ (N ) or, simply by plotting a histogram. For example, suppose that n = m = 10, X = 8 and Y = 6. The R code is: x <- 8; y <- 6 n <- 10; m <- 10 B <- 10000 p1 <- rbeta(B,x+1,n-x+1) p2 <- rbeta(B,y+1,m-y+1) delta <- p2 - p1 print(mean(delta)) left <- quantile(delta,.025) right <- quantile(delta,.975) print(c(left,right)) I found that Æ and a 95 per cent posterior interval of (-0.52,0.20). The posterior density can be estimated from a histogram of the simulated values as shown in Figure 25.1. Suppose we conduct an experiment by giving rats one of ten possible doses of a drug, denoted by x1 < x2 < : : : < x10 . For each dose level xi we use n rats and we observe Yi , the number that survive. Thus we have ten independent binomials Yi Binomial(n; pi ). Suppose we know from biological considerations that higher doses should have higher probability of death. Thus, p1 p2 p10 . Suppose we want to estimate the dose at which the animals have a 50 per cent chance of dying. This is called the LD50. Formally, Æ = xj where j = minfi : pi :50g: Example 25.4 (Dose Response) Notice that Æ is implicitly a (complicated) function of p1 ; : : : ; p10 so we can write Æ = g (p1 ; : : : ; p10 ) for some g . This just means that if we know (p1 ; : : : ; p10 ) then we can nd Æ . The posterior 511 1.0 0.5 0.0 f(delta | data) 1.5 2.0 25.2 Basic Monte Carlo Integration −0.8 −0.6 −0.4 −0.2 0.0 delta = p2 − p1 FIGURE 25.1. Posterior of Æ from simulation. 0.2 0.4 0.6 512 25. Simulation Methods mean of Æ is Z Z Z g(p ; : : : ; p )f (p ; : : : ; p jY ; : : : ; Y )dp dp : : : dp : A 1 10 1 10 1 10 1 2 10 The integral is over the region A = f(p1 ; : : : ; p10 ) : p1 p10 g: Similarly, the posterior cdf of Æ is F (cjY1; : : : ; Y10 ) = P (Æ cjYZ1 ; : : : ; Y10 ) Z Z f (p ; : : : ; p jY ; : : : ; Y )dp dp : : : dp = B 1 10 1 10 1 2 where B=A \ f(p ; : : : ; p 1 10 ) : g (p1; : : : ; p10 ) cg: We would need to do a 10 dimensional integral over a resticted region A. Instead, let's use simulation. Let us take a at prior truncated over A. Except for the truncation, each Pi has once again a Beta distribution. To draw from the posterior we do the following steps: (1) Draw Pi Beta(Yi + 1; n Yi + 1); i = 1; : : : ; 10. (2) If P1 P2 P10 keep this draw. Otherwise, throw it away and draw again until you get one you can keep. (3) Let Æ = xj where j = minfi : Pi > :50g: We repeat this N times to get Æ (1) ; : : : ; Æ (N ) and take E (Æ jY1; : : : ; Y10 ) 1 N X i Æ (i) : 10 25.2 Basic Monte Carlo Integration 513 Æ is a discrete variable. We can estimate its probability mass function by P (Æ = xj jY1 ; : : : ; Y10 ) 1 N N X i=1 I (Æ (i) = j ): For example, suppose that x = (1; 2; : : : ; 10). Figure 25.2 shows some data with n = 15 animals at each dose. The R code is a bit tricky. ## assume x and y are given n <- rep(15,10) B <- 10000 p <- matrix(0,B,10) check <- rep(0,B) for(i in 1:10){ p[,i] <- rbeta(B, y[i]+1, n[i]-y[i]+1) } for(i in 1:B){ check[i] <- min(diff(p[i,])) } good <- (1:B)[check >= 0] p <- p[good,] B <- nrow(p) delta <- rep(0,B) for(i in 1:B){ temp <- cumsum(p[i,]) j <- (1:10)[temp > .5] j <- min(j) delta[i] <- x[j] } print(mean(delta)) left <- quantile(delta,.025) right <- quantile(delta,.975) print(c(left,right)) 1.0 25. Simulation Methods 0.8 0.8 t(p) 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 proportion who died 1.0 514 2 4 6 8 10 0.4 0.2 0.0 f(delta | data) 0.6 0.8 dose 3 4 5 LD50 FIGURE 25.2. Dose repoonse data. wpq6 m N 3zc N N Nuonhgeba5d0982174yxvskrtflji N 6 N N i N N pNg983eyvr w m qnha41bzxskcj N Nuod7520tfl 2 N Nq653xzvtj N N8 N m Nuphgd97yri wnoeb4skcfl N 01a2 Nd0 N96rt N wuqocl N N m Nnhgea7541xzvsj N N83y N pbkfi Nugt N N N po2bazxvk N N q607 N N Nnhed4315crl N8s myfji 9 w N Nut 1 Na 6 m Nhgd750 N Nqpoe92xvcrl w4 N N Nbskij N nzf N38y ua5i N N wqpoh6bvrt N Nngd9870321zyxskflj me4c qn6 N N wupohgba5d9821zxyvskcrtij N N e703l m 4f qn6a5st N N N wupohgeb94321d08zxvyckrfi m 7lj N m wuqpnogba651ed432h0987xzyvskcrtflji N 2 4 6 x 8 10 25.3 Importance Sampling 515 The posterior draws for p1 ; : : : ; p10 are shown in the second panel in the gure. We nd that that Æ = 4:04 with a 95 per cent interval of (3,5). The third panel shows the posterior for Æ which is necessarily a discrete distribution. 25.3 Importance Sampling R Consider the integral I = h(x)f (x)dx where f is a probability density. The basic Monte Carlo method involves sampling from f . However, there are cases where we may not know how to sample from f . For example, in Bayesian inference, the posterior density density is is obtained by multiplying the likelihood L() times the prior f (). There is no guarantee that f (jx) will be a known distribution like a Normal or Gamma or whatever. Importance sampling is a generalization of basic Monte Carlo which overcomes this problem. Let g be a probability density that we know how to simulate from. Then I= Z h(x)f (x)dx = Z h(x)f (x) g (x)dx = Eg (Y ) g (x) where Y = h(X )f (X )=g (X ) and the expectation Eg (Y ) is with respect to g . We can simulate X1 ; : : : ; XN g and estimate I by 1 X h(Xi )f (Xi ) 1X Yi = : Ib = N i N i g (Xi ) This is called importance sampling. By the law of large nump bers, Ib ! I . However, there is a catch. It's possible that Ib might have an innite standard error. To see why, recall that I is the mean of w(x) = h(x)f (x)=g (x). The second moment of this quantity is Eg (w (X )) = 2 Z h(x)f (x) g (x) 2 g (x)dx = Z h2 (x)f 2 (x) dx: g (x) 516 25. Simulation Methods If g has thinner tails than f , then this integral might be innite. To avoid this, a basic rule in importance sampling is to sample from a density g with thicker tails than f . Also, suppose that g (x) is small over some set A where f (x) is large. Again, the ratio of f=g could be large leading to a large variance. This implies that we should choose g to be similar in shape to f . In summary, a good choice for an importance sampling density g should be similar to f but with thicker tails. In fact, we can say what the optimal choice of g is. Theorem 25.5 is The choice of g that minimizes the variance of Ib g (x) = jh(x)jf (x) : jh(s)jf (s)ds R PROOF. The variance of w = fh=g is Eg (w ) (E (w )) = 2 2 Z 2 = = w (x)g (x)dx Z 2 Z Z h2 (x)f 2 (x) g (x)dx g 2 (x) h2 (x)f 2 (x) g (x)dx g 2 (x) 2 w(x)g (x)dx Z Z 2 h(x)f (x) g (x)dx g (x) h(x)f (x)dx The second integral does not depend on g so we only need to minimize the rst integral. Now, from Jensen's inequality we have Eg (W ) (Eg (jW j)) = 2 2 Z 2 2 jh(x)jf (x)dx : This establishes a lower bound on Eg (W 2 ). However, Eg (W 2 ) equals this lower bound which proves the claim. This theorem is interesting but it is only of theoretical interest. If we did not know how to sample from f then it is unlikely : 25.3 Importance Sampling 517 R that we could sample from jh(x)jf (x)= jh(s)jf (s)ds. In practice, we simply try to nd a thick tailed distribution g which is similar to f jhj. Let's Restimate I = P (Z > 3) = :0013 where Z N (0; 1). Write I = h(x)f (x)dx where f (x) is the standard Normal density and h(x) = 1 if x > 3 andP0 otherwise. The basic Monte Carlo estimator is Ib = N 1 i h(Xi ). Using N = 100 we nd (from simulating many times) that E (Ib) = :0015 and V ar(Ib) = :0039. Notice that most observations are wasted in the sense that most are not near the right tail. We will estimate this with importance sampling taking g to be a Normal(4,1) density. We draw values from g and the P 1 b estimate is now I = N i f (Xi )h(Xi )=g (Xi ). In this case we b nd that E (I ) = :0011 and V ar(Ib) = :0002: We have reduced the standard deviation by a factor of 20. Example 25.6 (Tail Probability) Suppose we have measurements X1 ; : : : ; Xn of some physical quantity . We might model this as Xi = + i : Example 25.7 (Measurement Model With Outliers) If we assume that i N (0; 1) then Xi N (i ; 1). However, when taking measurements, it is often the case that we get the occasional wild obvervation, or outlier. This suggests that a Normal might be a poor model since Normals have thin tails which implies that extreme observations are rare. One way to improve the model is to use a density for i with a thicker tail, for example, a t-distribution with degrees of freedom which has the form ( +1)=2 +1 2 x 1 2 : t(x) = 1 + 2 Smaller values of correspond to thicker tails. For the sake of illustration we will take = 3. Suppose we observe n Xi = + i , i = 1; : : : ; n where i has a t distribution with = 3. We will 518 25. Simulation Methods take a at prior on . The likelihood is L() = and the posterior mean of is = R R Qn i=1 t(Xi ) L()d L()d : We can estimate the top and bottom integral using importance sampling. We draw 1 ; : : : ; N g and then PN j L(j ) j =1 g(j ) P N L(j ) : 1 j =1 g(j ) N 1 N To illustrate the idea, I drew n = 2 observations. The posterior mean (ccomputed numerically) is -0.54. Using a Normal importance sampler g yields an estimate of -0.74. Using a Cauchy (tdistribution with 1 degree of freedom) importance sampler yields an estimate of -0.53. 25.4 MCMC Part I: The Metropolis-Hastings Algorithm R Consider again the problem of estimating the integral I = h(x)f (x)dx. In this chapter we introduce Markov chain Monte Carlo (MCMC) methods. The idea is to constuct a Markov chain X1 ; X2 ; : : : ; whose stationary distribution is f . Under certain conditions it will then follow that 1 N N X i=1 p h(Xi ) ! Ef (h(X )) = I: This works because there is a law of large numbers for Markov chains called the ergodic theorem. The Metropolis-Hastings algorithm is a specic MCMC method that works as follows. Let q (y jx) be an arbitrary, friendly distribution (i.e. we know how to sample from q (y jx)). The conditional density q (y jx) is called the proposal distribution. 25.4 MCMC Part I: The Metropolis-Hastings Algorithm 519 The Metropolis-Hastings algorithm creates a sequence of observations X0 ; X1 ; : : : ; as follows. Choose X0 arbitrarily. Suppose we have generated X0 ; X1 ; : : : ; Xi. To generate Xi+1 do the following: Metropolis-Hastings Algorithm. (1) Generate a proposal or candidate value Y (2) Evaluate r r(Xi; Y ) where q(yjXi). f (y ) q (xjy ) r(x; y ) = min ;1 : f (x) q (y jx) (3) Set Xi+1 = Y with probability r Xi with probability 1 r: REMARK 1: A simple way to execute step (3) is to generate U (0; 1). If U < r set Xi+1 = Y otherwise set Xi+1 = Xi . REMARK 2: A common choice for q (y jx) is N (x; b2 ) for some b > 0. This means that the proposal is draw from a Normal, centered at the current value. REMARK 3: If the proposal density q is symmetric, q (y jx) = q (xjy ), then r simplies to f (Y ) ;1 : r = min f (Xi ) The Normal proposal distribution mentioned in Remark 2 is an example of a symmetric proposal density. 520 25. Simulation Methods By construction, X0 ; X1 ; : : : is a Markov chain. But why does this Markov chain have f as its stationary distribution? Before we explain why, let us rst do an example. The Cauchy distribution has density 1 1 f (x) = : 1 + x2 Our goal is to simulate a Markov chain whose stationary distribution is f . As suggested in the remark above, we take q (y jx) to be a N (x; b2 ). So in this case, Example 25.8 f (y ) 1 + x2 r(x; y ) = min ; 1 = min ; 1 : f (x) 1 + y2 So the algorithm is to draw Y Xi+1 = N (Xi ; b ) and set 2 Y with probability r(Xi ; Y ) Xi with probability 1 r(Xi; Y ): The R code is very simple: metrop <- function(N,b){ x <- rep(0,N) for(i in 2:N){ y <- rnorm(1,x[i-1],b) r <- (1+x^2)/(1+y^2) u <- runif(1) if(u < r)x[i] <- y else x[i] <- x[i-1] } return(x) } The simulator requires a choice of b. Figure 25.3 shows three chains of length N = 1000 using b = :1, b = 1 and b = 10. The histograms of the simulated values are shown in Figure 25.4. Setting b = :1 forces the chain to take small steps. As a result, the chain doesn't \explore" much of the sample space. The histogram 25.4 MCMC Part I: The Metropolis-Hastings Algorithm 521 from the sample does not approximate the true density very well. Setting b = 10 causes the proposals to often be far in the tails, making r small and hence we reject the proposal and keep the chain at its current position. The result is that the chain \gets stuck" at the same place quite often. Again, this means that the histogram from the sample does not approximate the true density very well. The middle choice avoids these extremes and results in a Markov chain sample that better represents the density sooner. In summary, there are are tuning parameters and the eÆciency of the chain depends on these parameters. We'll discus this in more detail later. If the sample from the Markov chain starts to \look like" the target distribution f quickly, then we say that the chain is \mixing well." Constructing a chain that mixes well is somewhat of an art. Why It Works. Recall from the previous chapter that a distribution satises detailed balance for a Markov chain if pij i = pjij : We then showed that if satises detailed balance, then it is a stationary distribution for the chain. Because we are now dealing with continuous state Markov chains, we will change notation a little and write p(x; y ) for the probability of making a transition from x to y . Also, let's use f (x) instead of for a distribution.R In this new notation, f is a stationary distribution if f (x) = f (y )p(y; x)dy and detailed balance holds for f if f (x)p(x; y ) = f (y )p(y; x): Detailed balance implies that f is a stationary distribution since, if detailed balance holds, then Z f (y )p(y; x)dy = Z f (x)p(x; y )dy = f (x) Z p(x; y )dy = f (x) 25. Simulation Methods −1.0 0.0 1.0 2.0 522 200 400 600 800 1000 0 200 400 600 800 1000 0 200 400 600 800 1000 −8 −4 0 2 4 −2 0 2 4 0 FIGURE 25.3. Three Metropolis chains corresponding to b = :1, b = 1, b = 10. 523 0.0 0.1 0.2 0.3 0.4 0.5 25.4 MCMC Part I: The Metropolis-Hastings Algorithm 0 5 −5 0 5 −5 0 5 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2 0.3 0.4 0.5 −5 FIGURE 25.4. Histograms from the three chains. The true density is also plotted. 524 25. Simulation Methods R which shows that f (x) = f (y )p(y; x)dy as required. Our goal is to show that f satises detailed balance which will imply that f is a stationary distribution for the chain. Consider two points x and y . Either f (x)q (y jx) < f (y )q (xjy ) or f (x)q (y jx) > f (y )q (xjy ): We will ignore ties which we can do in the continuous setting. Without loss of generality, assume that f (x)q (y jx) > f (y )q (xjy ). This implies that r(x; y ) = f (y ) q (xjy ) f (x) q (y jx) and that r(y; x) = 1. Now p(x; y ) is the probability of jumping from x to y . This requires two things: (i) the proposal distribution must generate y and (ii) you must accept y . Thus, p(x; y ) = q (y jx)r(x; y ) = q (y jx) f (y ) q (xjy ) f (y ) = q (xjy ): f (x) q (y jx) f (x) Therefore, f (x)p(x; y ) = f (y )q (xjy ): (25.1) On the other hand, p(y; x) is the probability of jumping from y to x. This requires two things: (i) the proposal distribution must generate x and (ii) you must accept x. This occurs with probability p(y; x) = q (xjy )r(y; x) = q (xjy ). Hence, f (y )p(y; x) = f (y )q (xjy ): (25.2) Comparing (25.1) and (25.2), we see that we have shown that detailed balance holds. 25.5 MCMC Part II: Dierent Flavors There are dierent types of MCMC algorithm. Here we will consider a few of the most popular versions. 25.5 MCMC Part II: Dierent Flavors 525 In the previous section we considered drawing a proposal Y of the form Random-walk-Metropolis-Hastings. Y = Xi + i where i comes from some distribution with density g . In other words, q (y jx) = g (y x). We saw that in this case, f (y ) r(x; y ) = min 1; : f (x) This is called a random-walk-Metropolis-Hastings method. The reason for the name is that, if we did not do the acceptreject step, we would be simulating a random walk. The most common choice for g is a N (0; b2 ). The hard part is choosng b so that the chain mixes well. A good rule of thumb is: choose b so that you accept the proposals about 50 per cent of the time. Warning: This method doesn't make sense unless X takes values on the whole real line. If X is restricted to some interval then it is best to transform X , say, Y = m(X ) say, where Y takes values on the whole real line. For example, if X 2 (0; 1) then you might take Y = log X and then simulate the distribution for Y instead of X . This is like an importancesampling version of MCMC. In this we draw the proposal from a xed distribution g . Generally, g is chosen to be an approximation to f . The acceptance probability becomes Independence-Metropolis-Hastings. f (y ) g (x) r(x; y ) = min 1; : f (x) g (y ) The two previous methods can be easily adapted, in principle, to work in higher dimensions. In practice, Gibbs Sampling. 526 25. Simulation Methods tuning the chains to make them mix well is hard. Gibbs sampling is a way to turn a high-dimensional problem into several one dimensional problems. Here's how it works for a bivariate problem. Suppose that (X; Y ) has density fX;Y (x; y ). First, suppose that it is possible to simulate from the conditional distributions fX jY (xjy ) and fY jX (y jx). Let (X0 ; Y0) be starting values. Asume we have drawn (X0 ; Y0 ); : : : ; (Xn; Yn). Then the Gibbs sampling algorithm for getting (Xn+1 ; Yn+1 ) is: The n + 1 step in Gibbs sampling: Xn+1 Yn+1 fX jY (xjYn) fY jX (yjXn +1 ) This generalizes in the obvious way to higher dimensions. Gibbs sampling is very useful for a class of models called hierarchical models. Here is a simple case. Suppose we draw a sample of k cities. From each city we draw ni people and observe how many people Yi have a disease. Thus, Yi Binomial(ni ; pi ). We are allowing for dierent disease rates in dierent cities. We can also think of the p0i s as random draws from some distribution F . We can write this model in the following way: Example 25.9 (Normal Hierarchical Model) Pi Yi jPi = pi F Binomial(ni; pi): We are interested in estimating the p0i s and the overall disease R rate pdF (p). To proceed, it will simplify matters if we make some transformations that allow us to use some Normal approxmations. Let 25.5 MCMC Part II: Dierent Flavors 527 p pbi = Yi =ni . Recall that pbi N (pi ; si) where si = pbi (1 pbi )=ni . Let i = log(pi =(1 pi )) and dene Zi bi = log(pbi =(1 pbi )). By the delta method, bi N ( i ; i ) 2 where i2 = 1=(npbi(1 pbi )). Experience shows that the Normal approximation for is more accurate than the Normal approximation for p so we shall work with . We shall treat i as known. Furthermore, we shall take the distribution of the i0 s to be Normal. The hierarchical model is now Zi j N (; ) N ( i ; i ): 2 i 2 i As yet another simplication we take = 1. The unknown parameter are = (; 1 ; : : : ; k ). The likelihood function is L() / / Y i Y i f ( i j) exp Y i 1 ( 2 f (Zi j ) i ) 2 exp 1 (Z 2i2 i i) 2 : If we use the prior f () / 1 then the posterior is proportional to the likelihood. To use Gibbs sampling, we need to nd the conditional distribution of each parameter conditional on all the others. Let us begin by nding f (jrest) where \rest" refers to all the other variables. We can throw away any terms that don't involve . Thus, Y 1 2 f (jrest) / exp ( ) 2 i i k 2 / exp 2 ( b) where 1X b= : k i i 528 25. Simulation Methods Hence we see that jrest N (b; 1=k). Next we will nd f ( jrest). Again, we can throw away any terms not involving i leaving us with 1 1 2 2 f ( i jrest) / exp ( ) exp (Z i) 2 i 2i2 i 1 2 / exp 2d2 ( i ei) i where ei = Zi i2 + 1 and d2i = 1 1 + 2 1 + 12 i i and so i jrest N (ei ; di ). The Gibbs sampling algorithm then involves iterating the following steps N times: 2 draw draw 1 .. . draw k N (b; v ) N (e ; d ) 2 1 2 1 .. . N (ek ; d2k ): It is understood that at each step, the most recently drawn version of each variable is used. Let's now consider a numerical example. Suppose that there are k = 20 cities and we sample n = 20 people from each city. The R function for this problem is: gibbs.fun <- function(y,n,N){ k <- length(y) p.hat <- y/n Z <- log(p.hat/(1-p.hat)) sigma <- sqrt(1/(n*p.hat*(1-p.hat))) v <- sqrt(1/sum(1/sigma^2)) mu <- rep(0,N) psi <- matrix(0,N,k) for(i in 2:N){ 25.5 MCMC Part II: Dierent Flavors 529 ### draw mu given rest b <- v^2*sum(psi[i-1,]/sigma^2) mu[i] <- rnorm(1,b,v) ### draw psi given rest e <- (Z + mu[i]/sigma^2)/(1 + (1/sigma^2)) d <- sqrt(1/(1 + (1/sigma^2))) psi[i,] <- rnorm(k,e,d) } list(mu=mu,psi=psi) } After running the chain, we can convert each i back into pi by way of pi = e =(1 + e ). The raw proportions are shown in Figure 25.6. Figure 25.5 shows \trace plots" of the Markov chain for p1 and . (We should also look at the plots for p2 ; : : : ; p20 .) Figure 25.6 shows the posterior for based on the simulated values. The second panel of Figure 25.6 shows the raw proportions and the Bayes estimates. Note that the Bayes estimates are \shrunk" together. The parameter controls the amount of shrinkage. We set = 1 but in practice, we should treat as another unknown parameter and let the data determine how much shrinkage is needed. i i So far we assumed that we know how to draw samples from the conditionals fX jY (xjy ) and fY jX (y jx). If we don't know how, we can still use the Gibbs sampling algorithm by drawing each observation using a Metropolis-Hastings step. Let q be a proposal distribution for x and let qe be a proposal distribution for y . When we do a Metropolis step for X we treat Y as xed. Similarly, when we do a Metropolis step for Y we treat X as xed. Here are the steps: 25. Simulation Methods 0.6 0.4 p1 0.8 530 0 200 400 600 800 1000 800 1000 −0.2 −0.6 mu 0.2 Simulated values of p1 0 200 400 600 Simulated values of mu FIGURE 25.5. Simulated values of p1 and . 531 0 50 100 150 200 25.5 MCMC Part II: Dierent Flavors −0.6 −0.4 −0.2 0.0 0.2 0.4 0.0 1.0 2.0 3.0 mu 0.0 0.2 0.4 0.6 raw estimates and Bayes estimates FIGURE 25.6. Top panel: posterior histogram of . Lower panel: raw proptions and the Bayes posterior estimates. 0.8 1.0 532 25. Simulation Methods Metropolis within Gibbs: (1a) Draw a proposal Z q (z jXn ). (1b) Evaluate f (Z; Yn) q (Xn jZ ) r = min ;1 : f (Xn; Yn ) q (Z jXn ) (1c) Set Xn+1 = Z with probability r Xn with probability 1 r: (2a) Draw a proposal Z qe(z jYn ). (2b) Evaluate f (Xn+1; Z ) qe(YnjZ ) r = min ;1 : f (Xn+1 ; Yn) qe(Z jYn) (2c) Set Yn+1 = Z with probability r Yn with probability 1 r: Again, this generalizes to more than two dimensions. 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