Chapter 3 Prior Information

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Οὐλίξης Νικολαΐδης
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1 Chatr Pror Iorato Subjtv Dtrato o th Pror Dst Svral usul aroah a b us to tr th ror st Th ar th hstogra aroah th rlatv llhoo aroah athg a gv utoal or 4 CDF trato () Th hstogra aroah Dv th aratr sa to trvals tr th subjtv robablt o ah trval a th lot a robablt hstogra () Th rlatv llhoo aroah Coar th tutv llhoos o varous ots a rtl sthg a ror st ro ths tratos () Mathg a gv utoal or Assu that ( ) s o a gv utoal or a to th hoos th st o ths gv or whh ost losl aths ror bls (4) CDF trato Subjtvl tr svral α rtls lot th ots ( z ( α )α ) sth a sooth urv jog th whr ( z ( α )) α P

2 ot: Th hstogra a th rlatv llhoo aroahs ar th ost usul ot: Utra or urt ol o qut oorl trg th robablt strbutos Through th stu o Basa robustss w a allvat ths robl o-oratv Prors (a) Itrouto o-oratv ror s a ror whh otas o orato about al : ( ) ( ) ( ) s so ostat ala (8) us ( ) As ( ) ( ) ( ) s ot a ror st (ror ror) I ato th othr svr rts o th o-oratv ror s th la o vara ur trasorato al :

3 s th aratr w ar trst Suos ( ) ( ) t Th th ror or s ( ) ( ) ( ( )) Thror th ror or s ot o-oratv (b) o-oratv ror or loato a sal robls () oato robl Th st o s o or ( ) ( ) ( s ol o ) Th st s th sa to b a loato st al : Th lt Th st o s ( ) ( z ) ( ) z ( ) For a ostat th st Y Y th Y has

4 ( ) ( ) ( ) ( ) Furthr ot th Y has th st ( ) whh s tal to th st o ( ) strutur al (otu): ( ) Y 5 Th Y ( 5 ) ( ) 5 st ( ) ( ) ( ) Y has th Assu ( ) a ( ) ar th o-oratv rors o a rstvl t A b a st Th A ( ) ( ) A A A ( ) Q ( ) A A A Thus ( ) ( ) t () ( ) so ostat It s ovt to hoos th ostat to b ; that s () 4

5 5 () Sal robl Th st o s o or > Th st s th sa to b a sal st al 4: Th lt z z Th st o s t Y Y Th Y has th st uto whr Y has th st whh s tal to th st o strutur al 4: 5 Y

6 Th 5 Y Th st o Y s Y Assu a ar th o-oratv rors o a rstvl t A b a st Th A A A A A A A Q Thus t () () It s ovt to st () Th () ot: s a ror ror s

7 7 ot: Th a ult usg th o-oratv ror s th uquss For al loato robl a b ossbl hos Thr ar a vart o rt hos () o-oratv rors gral sttgs t I b th t Fshr s orato Jrs (9) roos th o-oratv ror [ ] I I [ ] t s a vtor o aratr th [ ] t I K whr I M O M M al 5: Th

8 8 I Thus th o-oratv ror s [ ] I al : Th 4 4 I

9 Mau tro Prors Motvato: S artal orato s avalabl t s srabl to a ror that s as o-oratv as ossbl Dto (tro srt as): Assu { K K } s srt a lt b a robablt o Th tro o s ot b ε ( ) ot: ε ( ) ( ) [ ( )] [ ( ( ))] I ( ) th ( ) [ ( )] s to b al 7: Th al 8: Th K { } ( ) ε { K } ( ) ε ( ) ( ) 9

10 Th ollowg rsult a b us to th au tro as so artal ror orato [ g ] K avalabl or al as g s ( ) [ g ( )] ( ) ; as as g [ ] ( ) ( ) [ g ( )] ( ) g ; I ( ) [ g ( )] I ( ) ( z ] ( z ] [ ] P ( z ) Iortat Rsult: Th ror satsg [ g ( )] K has au tro ( ) g ( ) g ( ) whr K a b tr b quatos [ g ( )] K al 9: Th { } g ( ) 5 K

11 whh s gotr strbuto 5 Thror 5 Dto (tro otuous as): Assu s otuous Th tro o s ε whr s th atural oratv ror Iortat Rsult: Th ror satsg [ ] g g K has au tro g g whr K a b tr b quatos

12 [ g ( )] K al : ( ) Θ a assu s loato aratr Also lt ( ) Assu w ow g ( ) ( ) Thus That s Th ( ) ( ) ( ) ( ) al : ( ) R Θ a assu s loato aratr Also lt ( ) Assu w ow g ( ) ( ) Th ( )

13 os ot st s os ot st al : R Θ a assu s loato aratr Also lt Assu w ow Var g g Th Thus Var Thror ot: Two ults ars trg to us th au tro aroah to tr a ror: th to us a o-oratv ror th rvato o os ot st

14 4 Usg th Margal Dstrbuto (I) Itrouto Rall th arg st o s Θ Θ ( ) ( ) ( otuous ) ( ) ( ) ( srt ) al : ( ) ( ) ( ) Th th argal st o s ( ) ( ) ( ) ( ) ( ) ot: ( ) s sots all th rtv strbuto or t ( ) w b th argal st ur th orrt ror a ( ) b th argal st ur th wrog ror Th th statst obta ro th ata shoul b los to th sa 4

15 statst bas o ( ) ot o w ( ) b th o o ( ) w ( ) For al lt a w b th o o Itutvl th obsrv ata shoul b arou ot w To th orrt or ssbl ror tvl o oul rstrt th ho o th rors to so lass Th bas o so rtra th bst ror oul b ou Svral lasss o rors ar rqutl us Prors o a gv utoal or: Γ { : ( ) g ( ) Λ} whr Λ s so st a s all a hr-aratr o th ror al : Γ ( ) { : ( ) > > } s th hr-aratr Prors o a gv utoal or: Γ : K t ( ) ( ) s a st ( ) al 4: K ( ) 5

16 s a ow ostat Γ : ( ) ( ) ( ) < < > Prors los to a lt ror: A ror los to a ssbl ror woul also b rasoabl For al ε otaato lass s Γ { : ( ε ) ( ) εq( ) q } whr s a lass o ossbl otaatos a q ( ) s so st uto or (II) Pror slto Thr ar svral aroahs to slt a ssbl ror Th ar () th M-II aroah () th ot aroah () th sta aroah () M-II aroah t Γ b a lass o rors ur osrato M-II (au llhoo-t II) s to Γ satsg su [ ] Γ al 5:

17 7 K Th Th M-II tho s to a azg Thus [ ] { } a s s Thror { } a s al : { } : q q Γ ε ε Th [ ] [ ] [ ] q q q ε ε ε ε ε ε

18 ow M-II ror s to q whh azs q ( ) I s th lass o all ossbl strbutos a azs ( ) t ( ) δ b th strbuto wth ( ) P (all ass at ) S th q ( ) q( ) ( ) q( ) ( ) q( ) ( ) ( ) δ ( ) δ ( ) ( ε ) ( ) εδ ( ) () Mot aroah t ( ) a ( ) vara o wth rst to ( ) b th otoal a o Also lt a b th ow argal a o vara o wth rst to ( ) Th th ollowg quatos a b us to obta th ot o th ror st suh as ( ) 8 a [ ] ( ) ( Y ) : ( [ ]) Q [( ) ] [ ( )] ( ) [ ] ( QVar ( ) [ Var ( Y )] ( ( Y ) ( )) )

19 O sal al s ( ) ( ) al 7: ( ) ( ) ( ) Suos w ow S ( ) ( ) th Thus ( ) ( ) s th arorat ror () Dsta aroah t ( ) b th argal st stat o obta ro th ata Also lt ( ) ( ) ( ) Θ b th argal st wh th bst ror s ou Th w tr to to z 9

20 ( ) ( otuous ) ( srt ) ot: ( ) ( ) ( ) 5 Hrarhal Prors Hrarhal ror s all a ultstag ror t th ror lass Γ whr ( ) { ( ) } : Λ s o a gv utoal or Th so stag s to osr ( ) o th hr-aratr Th so stag ror ( ) s also all a hr-ror ot: ( ) ( ) ( ) Λ

21 al 8: K : > < < Γ α α α γ α γ z z z IG Z IG Q al 9: K : > Γ γ Th

22 I t t Σ Σ 5 t strbuto wth Multvarat K α γ γ γ γ γ γ γ γ γ

23 s γ s th st o IG a or a ultvarat-t strbuto Y wth aratrs [ ] Σ t α has th st [ ] t t Σ Σ α α γ α γ K Crtss I Objtvt Bo (98): t s ossbl all to stgush btw ol assutos a th ror strbuto o th aratrs II Msus o ror strbuto Qusto: Wh rt rasoabl rors l substatall rt

24 aswrs a t b rght to stat that thr s a sgl aswr? Woul t ot b bttr to at that thr s st urtat wth th oluso g o th ror bls? Aswr: I rortg olusos ro Basa aalss th ror (also ata a loss) shoul b rort saratl orr to allow othrs to valuat th rasoablss o th subjtv uts III Robustss Qusto: Slght hags th ror strbuto ght aus sgat hags th so Aswr: Through robust Basa thoo a ho o robust rors or a b ru 7 ral Bas Aalss Motvato: For ε ε Th th M stat s ( ) K (hghr soal ol) I th M stat s 4

25 (lowr soal ol) Qusto to as: whh ol s bttr (lowr or hghr)? Aswr: ral Bas aalss s artular srabl ths (a) Itrouto stuato ral Bas tho rovs a oros btw th ol whr ar oltl urlat (hghr soal ol) a that whr all th ar assu to b qual (lowr soal ol) Thr ar two ts o ral Bas thos O s aratr ral Bas (PB) a th othr s oaratr ral Bas (PB) Paratr ral Bas: th ror ( ) s so aratr lass wth uow hraratrs oaratr ral Bas: o tall assu ol that ar ro so ror ( ) al : t ( ) ( ) ( ) 5

26 whr a ar uow hraratrs Two rt was to arr out ral Bas aalss ar statg th ror or ostror b ata rst th us ( ) or ( ) to arr out th staar Bas aalss g th Bas rul tr o uow ror a us th ata to stat th Bas rul (b) Paratr ral Bas or oral a al 5 (otu): Th K ( ) ( ) ( ) Th ror usg M-II tho s whr ( ) ( ) ( ) ( ) ( a { s } ) a a{ s } Th ostror strbuto or s ( ) Th th aratr stat usg Bas rul ur squar-loss uto s th ostror a

27 7 B B B whr B Furthr th ostror vara s B V ot: B B Morrs (98 JASA) suggst B a th th ral Bas stat s B B V B B B B B B whr a ar th stats obta st Th stat ostror strbuto s B B V ot: B B

28 8 Th % α HPD rbl st or s B B B B B V z V z C α α a Th % α HPD rbl st or s : α χ B B B V C al : [ ] [ ] t Y Y Y Y ε ε K Th S Th argal strbuto s th [ ] [ ] t Σ K whr Σ M O M M M O M M M

29 9 a zg also z t Σ Thus t t Σ Σ whr Σ M O M M S volvs a volvs w hav to us tratv sh to solv Th ostror strbuto or Y s Slar to th rvous al th ral Bas stats ar B B l C C l V C whr l l C

30 a [ ] Σ t t l () o-aratr ral Bas aalss al : P K ar ro a oo ror Ur squar loss th stat s th ostror a!! δ Furthr w a stat th argal strbuto b th ral

31 strbuto I ( ) ( ) whr I ( ) as as Thus th ral Bas stat or s δ B ( ) ( ) I ( j ) j j I ( ) j

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