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1、6HypothesisTestingandModelSelectionForBayesians,modelselectionandmodelcriticismareextremelyimportantinferenceproblems.Sometimesthesetendtobecomemuchmorecomplicatedthanestimationproblems.Inthischapter,someoftheseissueswillbedis•cussedindetail.However,allmodelsandh
2、ypothesesconsideredherearelow-dimensionalbecausehigh-dimensionalmodelsneedadifferentapproach.TheBayesiansolutionswillbecomparedandcontrastedwiththecorrespondingproceduresofclassicalstatisticswheneverappropriate.Someofthediscussioninthischapteristechnicalanditwill
3、notbeusedintherestofthebook.Thosesectionsthatareverytechnical(orotherwisecanbeomittedatfirstreading)areindicatedappropriately.TheseincludeSections6.3.4,6.4,6.5,and6.7.InSections6.2and6.3,wecomparefrequentistandBayesianapproachestohypothesistesting.Wedothesameinan
4、asymptoticframeworkinSection6.4.RecentlydevelopedmethodologiessuchastheBayesianP-valueandsomenon-subjectiveBayesfactorsarediscussedinSections6.5and6.7.6.1PreliminariesFirst,letusrecallsomenotationfromChapter2andalsoletusintroducesomespecificnotationforthediscussi
5、onthatfollows.SupposeXhavingdensityf{x6)isobserved,with6beinganunknownel•ementoftheparameterspace0.SupposethatweareinterestedincomparingtwomodelsMQandMi,whicharegivenbyMQ:Xhasdensityf{x6)where6eOQ;Ml:Xhasdensityf{xe)where6eOi.(6.1)Fori=0,1letgi{0)bethepriorden
6、sityof0^conditionalonMibeingthetruemodel.Then,tocomparemodelsMQandMionthebasisofarandomsamplex={xi,...,Xn)onewouldusetheBayesfactor1606HypothesisTestingandModelSelectionBoi(x)=^,(6.2)mi(x)wherem,(x)=/f{-Ke)gi{e)de,i=0,l.(6.3)J0iWealsousethenotationBF^ifortheBaye
7、sfactor.RecallfromChapter2thattheBayesfactoristheratioofposterioroddsratioofthehypothesestothecorrespondingprioroddsratio.Therefore,ifthepriorprobabilitiesofthehypotheses,TTQ=P^'iMo)=P'^(Oo)andTTI-P^(Mi)=P^(0i)=1-TTQarespecified,thenasin(2.17),P(Mo
8、x)=11+^-^BoY(^
9、)}•(6-4)Thus,ifconditionalpriordensities^oandgcanbespecified,oneshouldsim•plyusetheBayesfactorPQIfoi*modelselection.If,furtherTTQisalsospecified,theposterioro
