Bayesian Analysis of Simple Random Densities

Transcription

1 Ope Joual of Statstcs Pulshed Ole ugust 4 ScRes Bayesa alyss of Smple Radom Destes Paulo C Maques F Calos de B Peea Depatameto de Estatístca Isttuto de Matemátca e Estatístca vesdade de São Paulo São Paulo Basl Emal: pmaques@meusp cpeea@meusp Receved 3 Jue 4; evsed 5 July 4; accepted July 4 Copyght 4 y authos ad Scetfc Reseach Pulshg Ic Ths wo s lcesed ude the Ceatve Commos ttuto Iteatoal Lcese (CC BY stact tactale opaametc po ove destes s toduced whch s closed ude samplg ad exhts pope posteo asymptotcs Keywods Bayesa opaametcs Bayesa Desty Estmato Radom Destes Radom Pattos Stochastc Smulatos Smoothg Itoducto The ealy 97 s wtessed Bayesa feece gog opaametc wth the toducto of statstcal models wth fte dmesoal paamete spaces The most cospcuous of these models s the Dchlet pocess [] whch s a po o the class of all poalty measues ove a gve sample space that tades geat aalytcal tactalty fo a educed suppot: as show y Blacwell [] ts ealzatos ae almost suely dscete poalty measues The posteo expectato of a Dchlet pocess s a poalty measue that gves postve mass to each oseved value the sample mag the pla Dchlet pocess usutale to hadle feetal polems such as desty estmato May extesos ad alteatves to the Dchlet pocess have ee poposed [3] I ths pape we costuct a po dstuto ove the class of destes wth espect to Leesgue measue Gve a patto sutevals of a ouded teval of the eal le we defe a adom desty whose ealzatos have a costat value o each suteval of the patto The dstuto of the values of the adom desty o each suteval s specfed y tasfomg ad codtog a multvaate omal dstuto Ou smple adom desty s the fte dmesoal aalogue of the stochastc pocesses toduced y Thou [4] ad Le [5] Computatos wth these stochastc pocesses volve a tactale omalzato costat ad ae estcted to values of the adom desty o a fte ume of ataly chose doma How to cte ths pape: Maques ad Peea C (4 Bayesa alyss of Smple Radom Destes Ope Joual of Statstcs

2 Maques C Peea pots demadg some d of tepolato of the esults The fte dmesoalty of ou adom desty maes ou computatos moe dect ad taspaet ad gves us smple statemets ad poofs outle of the pape s as follows I Secto we gve the fomal defto of a smple adom desty I Secto 3 we pove that the dstuto of a smple adom desty s closed ude samplg The esults of the smulatos Secto 4 depct the asymptotc ehavo of the posteo dstuto We exted the model heachcally Secto 5 to deal wth adom pattos lthough the usual Bayes estmate of a smple adom desty s a dscotuous desty Secto 6 we compute smooth estmates solvg a decso polem whch the states of atue ae ealzatos of the smple adom desty ad the actos ae smooth destes of a sutale class ddtoal popostos ad poofs of all the esults the pape ae gve Secto 7 Smple Radom Destes Let ( Ω F P e the poalty space fom whch we duce the dstutos of all adom ojects cosdeed the pape Fo some tege let e the suset of vectos of wth postve compoets Wte R fo the Boel sgma-feld of Let λ deote Leesgue measue ove ( R We omt the dexes whe The compoets of a vecto v ae wtte as v v Suppose that we have ee gve a teval [ a ] ad a set of eal umes { t t t } such that a t < t < < t ducg a patto of [ a ] to the sutevals [ at [ t t [ t t [ t ] The class of smple destes wth espect to ths patto cossts of the oegatve smple fuctos whch have a costat value o each suteval ad tegate to oe Let d t t fo ad defe the map S : y S ( u du Each smple desty f : wth ths class ca e epeseted as whch [ f x hi x t t h h h s such that each h ad S ( h The h s ae the heghts of the steps of the smple desty f Fom ow o let { v : dv dv } fo ote that y the defto of the d s gve aove t follows that f Moeove defe the pojecto o the fst coodates π : y π ( v v v ( v v Fo a omal adom vecto Z ( Z Z wth mea m ad covaace matx Σ deote y L ( m Σ the dstuto of the logomal adom Z Z e e If Σ s osgula t s easy to show that has a desty vecto exp ( log f u π Σ u u m Σ ( log u m I ( u whch Σ s the detemat of Σ ad we have toduced the otatos log u ( log u log u ad m ( m m We defe a adom desty whose ealzatos ae smple destes wth espect to the patto duced y specfyg the dstuto of the adom vecto of ts steps heghts Ifomally the steps heghts wll have the dstuto of a logomal adom vecto gve that S ( The fomal defto of the adom desty s gve tems of a veso of the codtoal dstuto of gve S ( ad the expesso of ts codtoal desty wth espect to a domatg measue Howeve we ae outsde the elemetay case whch the jot dstuto s domated y a poduct measue I fact we have Poposto 7 a smple poof that Leesgue measue λ ad the jot dstuto of ad S ( ae mutually sgula sutale famly of measues that domate the codtoal dstuto of gve S ( fo each value of S ( s desced the followg lemma Lemma Let τ : R e defed y τ ( d λ ( π ( fo The each τ s a measue ove ( R The poof of Lemma s gve Secto 7 Fgue gves a smple geometc tepetato of the measues τ whe the udelyg patto s fomed y thee sutevals The followg esult s the ass fo the fomal defto of the adom desty 378

3 Maques C Peea Fgue Geometcal tepetato of the measues τ of Lemma fo > the patcula case whe 3 π The value of τ s the aea of the pojecto multpled y d 3 e the famly of measues ove R defed o Lemma The we have that : S R defed y Theoem Let L ( m Σ wth osgula Σ ad let { τ } ( ( u ( f ( I ( u d S u f τ s a egula veso of the codtoal dstuto of gve S ( Moeove ( S f f u I u d u S τ whch S fo each > The ecessay lemmata ad the poof of Theoem ae gve Secto 7 The followg defto of the adom desty uses the specfc veso of the codtoal dstuto costucted Theoem L m Σ wth osgula Σ We say that the map ϕ : Ω defed y Defto 3 Let ( x H I [ ( x ϕ ω ω t t ae the adom heghts of the steps of ϕ wth dst- R ad S ( s the egula veso of the codtoal otaed Theoem Hece fo evey R we have s a smple adom desty whch H ( H H uto gve y H ( ( S fo dstuto of gve S ( f ( h H ( I d h τ h f ( S whch τ( d λ ( π ( ad t holds that H ( We use the otato ϕ ( m Σ 3 Codtoal Model ow we model a set of asolutely cotuous osevales codtoally gve the value of a smple adom desty ϕ The followg lemma poved Secto 7 desces the codtoal model ad detemes the fom of the lelhood ϕ m Σ epeseted as Lemma 3 Cosde 379

4 Maques C Peea ( x H I [ ( x ϕ ω ω t t ad let the adom vaales X X e codtoally depedet ad detcally dstuted gve that H h wth dstuto X d H h f y λ y whch we have defed f ( y hi [ ( y Defe t t X ( X X ad let x ( x x The XH ( h λ almost suely [ H ] wth Rado-odym devatve whch c I [ t t ( x j d XH XH λ d xh f xh h c fo j The factozato cteo mples that c ( c c s a suffcet statstc fo ϕ That s ths codtoal model as oe should expect all the sample fomato s cotaed the coutgs of how may sample pots elog to each suteval of the patto duced y sg the otato of Lemma 3 ad defg c ( c c we ca pove that the po dstuto of ϕ s closed ude samplg * * Theoem 3 If ϕ ( m Σ the ϕ X x ( m Σ whch m mσ c Ths esult poved Secto 7 maes the smulatos of the po ad posteo dstutos essetally the * same the oly dffeece eg the computato of m 4 Stochastc Smulatos We summaze the dstuto of a smple adom desty ϕ ( m Σ ϕ( x ω H ( ω I [ ( x epeseted as two ways Fst motvated y the fact poved Poposto 75 that the t t po ad posteo expectatos ae pedctve destes we tae as a estmate the expectato of the steps heghts hˆ E [ H] E[ H ] Secod the ucetaty of ths estmate s assessed defg ( ˆ ˆ { < } B h h : max h h ad tag as a cedle set the B ( h ˆ wth the smallest such that P : H B( hˆ { } fo > ω ω γ whch γ ( s the cedlty level The Radom Wal Metopols algothm [6] s used to daw depedet ealzatos of the steps of ϕ as values of a Maov cha H The two summaes ae computed though egodc meas of ths cha Fo { } example the cedle set s detemed wth the help of the almost sue covegece of { } I ( ˆ H E I ˆ ˆ H P ω: H ω B h Bh Bh s fo the paametes appeag Defto 3 we tae ou expemets all the m s equal to oe ad the covaace matx Σ ( σ j s chose the followg way Gve some postve defte covaace fucto C : we duce Σ fom C defg t t j t t j σ j C fo j I ou examples we study the famly of Gaussa covaace fuctos defed y θ ( x y C ρθ xy ρe wth dspeso paamete ρ > ad scale paamete θ > Example 4 Let ϕ ( m Σ ad cosde the sample space [ ] wth { 98 99} Fo the sae of geealty we duce Σ fom the famly of Gaussa covaace fuctos wth fxed dspeso paamete ρ ut wth adom scale paamete Θ Y whch Y Gamma ( These choces guaatee that computatos wth Σ ae umecally stale I Fgue the summaes of the po dstuto of ϕ show that the value of ρ cotols the cocetato of the po Fxg ρ 5 ad 38

5 Maques C Peea geeatg data fom the mxtue Beta ( Beta ( Beta ( we have Fgue 3 the posteo summaes fo dffeet sample szes ote the cocetato of the posteo as we cease the sze of the samples Fgue Effect of the value of ρ o the cocetato of the po The cuves lac ae po expectatos ad the gay egos ae cedle sets wth cedlty level of 95% Fgue 3 Posteo summaes fo Example 4 O each gaph the lac smple desty s the estmate ˆϕ the lght gay ego s a cedle set wth cedlty level of 95% ad the da gay cuve s the data geeatg desty 38

6 Maques C Peea We oseve the same asymptotc ehavo of the posteo dstuto wth data comg fom a tagula dstuto ad a mxtue of omals (wth appopate tucato of the sample space 5 Radom Pattos Ifeetally we have a che costucto whe the defto of the smple adom desty volves a adom patto Ifomally we wat a model fo the adom desty whch the udelyg patto adapts tself accodg to the fomato cotaed the data We cosde a famly of ufom pattos of a gve teval [ ] a Each patto of ths famly wll e desced y a postve tege adom vaale K whch detemes the ume of sutevals the patto Sce the paamete ρ of the famly of Gaussa covaace fuctos used to duce Σ may have dffeet meags fo dffeet pattos we teat t as a postve adom vaale R Explctly we ae cosdeg the followg heachcal model: K ad R ae depedet Gve that K ad R ρ a duced y we choose the ufom patto of the teval [ ] a ( a ( ( a aa a a duce Σ ρθ fom the famly of Gaussa covaace fuctos ad tae the smple adom desty ϕ ( m Σ ρθ Fally the osevales ae modeled as Lemma 3 Ths heachy s summazed the followg gaph I the followg example stead of specfyg pos fo K ad R we defe the lelhood of K ad R y Lx( ρ fxkr ( x ρ whose fom s otaed Poposto 76 fd the maxmum ˆ ˆ ρ ag max L ρ ad use these values the deftos of the po detemg the posteo sum- ρ x maes as we dd Secto 4 Example 5 Wth a sample of sze geeated fom a Beta ( 4 dstuto we fd the maxmum of the lelhood of K ad R at ( ˆ ˆ ρ ( 943 I Fgue 4 we have the posteo summaes otaed usg these values the defto of the po Moeove the left gaph of Fgue 5 we have the dstuto fucto ˆF coespodg to the estmated posteo desty Fo the sae of compaso we plot the ght gaph of Fgue 5 some quatles of ths dstuto ˆF agast the quatles of the dstuto F fom whch we geeated the data 6 Smooth Estmates It s possle to go eyod the dscotuous destes otaed as estmates the last two sectos ad get smooth estmates of a smple adom desty ϕ solvg a Bayesa decso polem whch the states of atue ae the ealzatos of ϕ ad the actos ae smooth destes of a sutale class I vew of Theoem 3 t s eough to cosde the polem wthout data s efoe the sample space s the a whch s pattoed accodg to some Fo a desty f wth espect to Leesgue measue we teval [ ] deote ts f f dλ Poposto 6 Fo let g g e destes wth espect to Leesgue measue wth suppot [ a ] such that g < ad let D e the class of destes of the fom α g wth α fo L om y 38

7 Maques C Peea Fgue 4 Posteo summaes fo Example 5 The lac smple desty s the estmate ˆϕ the lght gay ego s a cedle set wth cedlty 95% ad the da gay cuve s the data geeatg desty Fgue 5 Example 5 O the left gaph the lac cuve s the estmated dstuto fucto ˆF ad the gay cuve s the data geeatg dstuto fucto F O the ght gaph we have the compaso of some of the quatles of ˆF ad F Let ad α ϕ m Σ ad defe S as the class of destes whch ae ealzatos of ϕ Defe the loss fucto L : S D y ( λ L s f s f s x f x d x The the Bayes decso s ˆ ϕ ˆ α g whch ˆ α mmze gloally the quadatc fom a Q αα M αj j j j α λ suject to the costats α fo ad Mj g x g j x d x a ϕ λ J g x E x d x a wth the deftos We use the esult of Poposto 6 poved Secto 7 choosg the g s sde a class of smooth destes that seve appoxmately as a ass to epeset ay cotuous desty wth the specfed suppot Fo the ext example suppose that the suppot of the destes s the teval [ ] Beste s Theoem (see [7] Theoem 6 states that the polyomal B x f x x ( appoxmates ufomly ay cotuous fucto f defed o [ ] whe Suppose that f s a desty 383

8 Maques C Peea If we defe fo ( ( Γ ( Γ Γ α f we ca ewte the appoxmatg polyomal as B ( x α g ( x Beta ( desty wth suppot [ ] whch g s a desty of a adom vaale Hece f we tae a suffcetly lage we expect that ay cotuous wll e easoaly appoxmated y a mxtue of these g s Example 6 Suppose that we have a sample of 5 data pots smulated fom a tucated expoetal dstuto whose desty s ( x e f ( x I [ ] ( x e Repeatg the aalyss made Example 5 we fd the maxmum of the lelhood of K ad R at ( ˆ ˆ ρ ( 9 86 The left gaph of Fgue 6 pesets the posteo summaes fte that we solved the polem of costaed optmzato Poposto 6 ad foud the esults show the ght gaph of Fgue 6 7 ddtoal Results ad Poofs I ths secto we peset some susday popostos ad gve poofs to all the esults stated the pape Poposto 7 Let L ( m Σ ad deote y S ( the jot dstuto of ad S ( The λ S ( Poof Defe the set { v : dv v } R The { } ( ( P ω: ( ω S ( ( ω P ω: d ω S ω S y defto of S O the othe had ote that λ ( sce ths s the ( c S mesoal hypeplae defed y the set Sce the esult follows -volume of the -d- Poof of Lemma Whe the esult s tval sce ths case mag τ a ull measue Suppose that > ad let g : e the fucto defed y Defe h y h ( y g( y g( v v v v dv d : y π The thee s a v Suppose that We wll show that π ( h ( such that y π ( v ( v v h( y g( y v v dv d fo evey R ad Fgue 6 Example 6 O the ght gaph the lac smple desty s the estmate ˆϕ ad the lght gay ego s a cedle set wth cedlty 95% O oth gaphs the da gay cuve s the data geeatg desty O the left gaph the lac smooth desty s the Bayes decso of Poposto 6 384

9 Sce v we have that ( dv v d mplyg that h that Maques C Peea h y v Sce v t follows fom the defto of the vese mage of y h ad theefoe we coclude that π ( h ( To pove the othe cluso suppose that y h ( v ad y the defto of h we have that v g( y y y dy d ad defe v h ( y Hece mplyg that v ecause dv Sce v y π v t follows that y π ( Theefoe h ( π ( Hece we have that τ d λ h ad the popetes of the vese mage of h ad the Leesgue measue etal that each τ s a measue ove ( R L m Σ Let ξ defed y Lemma 7 Let e a measue ove ( ad { } ( ξ λ u : us u the jot dstuto of ad S ( R Deote y S ( that S ( ξ wth Rado-odym devatve d dξ f S S( gve y f u f u I u S ( The we have whch u ad Poof Defe the fucto T : y ( T u us u ote that ξ λ T Defe the fucto ψ : ψ u f u I u wth u ad The dagam y commutes sce ψ ( T( u ψ ( u S u f u I u f u we have that S fo evey ( ( f ( u dλ u d T ψ T u λ T ( u ψ ( u ξ( u f ( u I ( u ξ( u u Fo evey { } { } P ω: ω S ω P ω: ω T S d d R whch the ffth equalty s otaed tasfomg y T u ad It follows that S ( ξ ad the Rado-odym devatve has the desed expesso Lemma 73 Let ξ e the measue defed o Lemma 7 ad let { τ } e the famly of measues defed o Lemma The fo evey measuale oegatve ψ : we have that ψ u d ξ u ψ u dτ u d λ whch u ad Poof Defe f : y f( u ( u u du Hece f s a dffeetale fucto whose vese s the dffeetale fucto g defed o Lemma The value of the Jacoa o the pot v s Jg ( v d Let R y ad defe h as Lemma Whe > we have aleady show the couse of the poof of Lemma that π ( h ( fo evey R Rememeg that y defto π h I y I ( g( y s fo the value of I ( g( y t follows that ( ow suppose that I ths case sce cosde two sucases: sce ad we coclude that π we have that I ( y I ( y π 385

10 Maques C Peea f ay of the y the I ( g( y g( y y y dy d othewse we have dy d < ad aga t happes that I ( g( y I ( y I ( g( y π Hece fo R ad B R we have that ξ( B λ { u : u S ( u B} I ( u IB( S ( u dλ ( u I ( g( y IB Jg ( y d λ ( y d I ( y I d ( B λ y π ( d λ y λ τ( λ B π B Theefoe we coclude that ths case also d d d whch y ad the thd equalty s otaed tasfomg y f ad the peultmate equalty s a cosequece of Toell s Theoem The esult follows fom the Poduct Measue Theoem ad Fu s Theoem (see [8] Theoems 6 ad 64 Lemma 74 Let L ( m Σ Let { τ } e the famly of measues defed o Lemma Let S ( e the dstuto of S ( The S ( λ wth Rado-odym devatve d dλ f S S( gve y Poof Let R S f f u I u d u S τ u ad Let ξ e the measue defed o Lemma 7 We have that { : } { : ( } ( f ( u I ( u d ( u S ( f ( u I ( u d τ d u λ ( ( P ω S ω P ω ω S ω ξ whch the peultmate equalty follows fom Lemma 7 ad the last equalty follows fom Lemma 73 Hece S ( λ ad the Rado-odym devatve has the desed expesso Poof of Theoem Let S ( e the jot dstuto of ad S ( ad let S ( e the S Fo R ad B R y the defto of codtoal dstuto we have that dstuto of ( B P{ S ( B} ( d S B S S B d S ( ( d λ S dλ whch we have used the Lez ule fo the Rado-odym devatves O the othe had y Lemmas 7 ad 73 we have that wth ( B f u I u d ξ u f u I u dτ u d λ S B B u ad Both expessos fo ( B S S ( ae compatle f dτ fs ( f u I u u 386

11 fo almost evey [ λ ] Theefoe we have that ( S odym devatve d S ( dτ f S ( ( gve y f ( u S fs ( as desed The fact that ( Poof of Lemma 3 Let τ fo almost evey f u I u > [ ] Maques C Peea λ wth Rado- follows mmedately S α h e the measues ove ( c R defed y α ( ( h λ ( x h d each h Let B B B wth B R fo By the hypothess of codtoal depedece ad Toell s Theoem we have that Hece XH ( h ( Bh ( B h f( x d ( x λ XH Xj H j B j j j j j B B j j f ( xj dλ( x hi [ ( x d t t j λ x c h d λ( x αh( B B ad α h agee o the π -system of poduct sets that geeate R Theefoe y Theoem 6 of [9] oth measues agee o the whole sgma-feld R It follows that XH ( h λ almost suely [ H ] ad the Rado-odym devatve has the desed expesso Poof of Theoem 3 By Bayes Theoem fo each R we have that c ( x C f ( xh d ( h C h d ( h HX XH H H d C h h h c H dτ ( dτ C c h f ( h I ( h d τ h f S whch we have used the expesso of the lelhood otaed Lemma 3 the Lez ule fo the Rado-odym devatves the expesso of d H dτ Defto 3 ad the costat C s such that x The emade of the poof eles o some matx algea Let I e the detty matx Sce HX y defto Σ s symmetc we have that I I ( Σ Σ Wte l log h Sce the scala l Σ m s equal to ts taspose that Defg d Σ c we have fo ΣΣ Σ Σ Σ Σ Theefoe we have that l m Σ l m l Σ l m Σ l m Σ m l m m l c * * h exp ( l m Σ ( l m exp ( d Σ l l Σ l m Σ l m Σ m C ( d Σ l l Σ l m Σ l m Σ m m Σ d d Σ d exp wth C exp ( ( m d d d d Σ m ( d Σ m m Σ d we have that Σ Σ Defe * m m d Sce the scala * * m m m m m d d d Σ Σ we have Σ Σ Σ Σ Hece we ota 387

12 Maques C Peea c * * h exp ( l m Σ ( l m C exp l Σ l m Σ l m Σ m * * C exp ( l m Σ ( l m * * * ( sg ths esult the expesso of HX togethe wth the expesso of f we have whch ( x C f * ( h I ( h ( h d τ HX C CC f S ad * L m Σ We coclude that gve that X x the vecto H has the dstuto of the heghts of the steps of ϕ ( m Σ as desed Poposto 75 Suppose that the adom vaales X X ae modeled accodg to Lemma 3 Deote y X the dstuto of X fo Fo coveece use the otatos X ( X X ad ( x x x The fo evey R we have X ( E ϕ( y dλ( y fo ; E d x ϕ y X x λ y X X as ( X Poof By Defto 3 we have whch h ad f ( y hi [ ( y t t Fo tem ote that f s a desty of the adom vecto E ( y E HI [ ( y ϕ f ( y d ( h t t H fo y I a aalogous mae we have E ϕ y X x f y d hx HX ( P{ X H } ( h d ( h ( f ( y dλ ( y d H ( h ( f ( y d H ( h dλ( y X X H H E ϕ( y d λ( y whch the fouth equalty follows fom Toell s Theoem Fo tem fo each P{ X X B} ( x ( B X X X x d O the othe had we have P X X B P X X BH B R { } { } ( x h d ( x h B X X H X H X H h d ( x h B X H ( d d B X H h hx x HX X ( ( d d d B f y λ y hx x HX X ( ( f( y d ( hx d ( y d ( x B λ HX X ( E ϕ( y X x dλ ( y d ( x B X we have 388

13 Maques C Peea whch the thd equalty follows fom the hypothess of codtoal depedece ad Theoem B6 of [9] the fouth equalty s a cosequece of Theoem 64 of [8] ad the sxth equalty s due to Toell s Theoem Compag oth expessos fo P{ X X B} we get the desed esult Poposto 76 Let K P K ove ( e the dstuto of K ad let R P R ove ( R e the dstuto of R Deote y KR the jot dstuto of K ad R whch y the depedece of K ad R s equal to the poduct measue K R ad let KRH e the jot dstuto of K R ad H I the heachcal model desced Secto 5 we have that XKR ( ρ λ almost suely KR wth Rado-odym devatve d XKR dλ ( x ρ fxkr ( x ρ fxh ( xh HKR ( h ρ d fo the f XH defed o Lemma 3 Poof Let R ad B R By the defto of codtoal dstuto we have { } XKR ( ρ KR ( ρ P X K R B d O the othe had y agumets smla to those used the poof of Poposto 75 we have P X K R B { } P{ X ( KR BH } XKRH ( ρ h d KRH ( ρ h B XH ( h d KRH ( ρ h B ( XH ( h d HKR ( h d KR ( B ρ ρ ( B ( B Compag oth expessos fo P{ X ( K R B} we have B ( fxh xh dλ x d HKR h ρ d KR ( ρ ( fxh xh HKR h ρ λ x KR ( ρ d d d ρ f xh d h ρ d λ x XKR XH HKR almost suely KR ad the esult follows Poof of Poposto 6 By Toell s Theoem the expected loss s ( ϕ λ ϕ λ a a whch we have defed the postve costat C E ϕ ( x dλ( x f a ( x α g ( x leadg us to E L f f x d x f x E x d x C By hypothess each f has the fom ( j ( a j a E L ϕ f αα g x g x dλ x α g x E ϕ x d λ x C j whch we have used the leaty of the tegal Theefoe mmzg the expected loss s the same as solvg the polem of costaed mmzato of the quadatc fom Q Fo the matx M ( M j ote that fo evey o ull ( y y y we have j j ( j j dλ a y My yym yy g x g x x j j ( j j d d a yg x y g x λ x yg x λ x a > j whch we have used the leaty of the tegal Theefoe the matx M s postve defte yeldg (see [] that the quadatc fom Q s covex ad the polem of costaed mmzato of Q has a sgle gloal 389

14 Maques C Peea soluto ( ˆ α ˆ α 8 Cocluso Sce the Bayes decso s the f that mmzes the expected loss the esult follows The adom desty cosdeed the pape ca e exteded to multvaate polems toducg aalogous pattos of d-dmesoal Eucldea space lso as a alteatve to the empcal appoach used Secto 5 we ca specfy full pos fo the hypepaametes lthough moe computatoally challegg ths choce defes a moe flexle model wth adom dmeso fo whch the desty estmates ae o loge smple destes Moe geeal adom pattos ca also e cosdeed Refeeces [] Feguso T (973 Bayesa alyss of Some opaametc Polems The als of Statstcs [] Blacwell D (973 Dsceteess of Feguso Selectos The als of Statstcs [3] Gosh JK ad Ramamooth RV ( Bayesa opaametcs Spge ew Yo [4] Thou D (986 Bayesa ppoach to Desty Estmato Bometa [5] Le PJ (988 The Logstc omal Dstuto fo Bayesa opaametc Pedctve Destes Joual of the meca Statstcal ssocato [6] Roet CP ad Casella G (4 Mote Calo Statstcal Methods d Edto Spge ew Yo [7] Bllgsley P (995 Poalty ad Measue 3d Edto Wley-Itescece ew Jesey [8] sh RB ( Poalty ad Measue Theoy 3d Edto Hacout/cademc Pess Massachusetts [9] Schevsh MJ (995 Theoy of Statstcs Spge ew Yo [] Bazaaa MS ad Shetty CM (6 olea Pogammg: Theoy ad lgothms 3d Edto Wley-Itescece ew Jesey 39

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