Bayesian computation with R-8.pdf

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1、8ModelComparison8.1IntroductionInthischapter,weillustratetheuseofRtocomparemodelsfromaBayesianperspective.WeintroducethenotionofaBayesfactorinthesettingwhereoneiscomparingtwohypothesesaboutaparameter.Inthesettingwhereoneistestinghypothesesaboutapopulationmean,weillustratethecomputa

2、tionofBayesfactorsinboththeone-sidedandtwo-sidedsettings.WethengeneralizetothesettingwhereoneiscomparingtwoBayesianmodels,eachconsistingofachoiceofpriorandsamplingdensity.Inthiscase,theBayesfactoristheratioofthemarginaldensitiesforthetwomodels.WeillustrateBayesfactorcomputationsint

3、woexamples.Intheanalysisofhittingdataforabaseballplayer,onewishestocomparea“consistent”modelwitha“streaky”modelwheretheprobabilityofasuccessmaychangeoveraseason.Inthesecondapplication,weillustratethecomputationofBayesfactorsagainstindependenceinatwo-waycontingencytable.8.2Compariso

4、nofHypothesesTointroduceBayesianmeasuresofevidence,supposeoneobservesYfromasamplingdistributionf(y

5、θ)andonewishestotestthehypothesesH0:θ∈Θ0,H1:θ∈Θ1,whereΘ0andΘ1formapartitionoftheparameterspace.Ifoneassignsaproperpriordensityg(θ),thenonecanjudgethetwohypothesesaprioribythepriorodds

6、ratioπ0P(θ∈Θ0)Θ0g(θ)dθ==.π1P(θ∈Θ1)Θg(θ)dθ1J.Albert,BayesianComputationwithR,UseR,DOI10.1007/978-0-387-92298-08,©SpringerScience+BusinessMedia,LLC20091828ModelComparisonAfterdataY=yareobserved,one’sbeliefsabouttheparameterareupdatedbytheposteriordensityg(θ

7、y)∝L(θ)g(θ),whereL(θ)ist

8、helikelihoodfunction.One’snewbeliefsaboutthetwohy-pothesesaresummarizedbytheposterioroddsratiop0P(θ∈Θ0

9、y)Θ0g(θ

10、y)dθ==.p1P(θ∈Θ1

11、y)Θg(θ

12、y)dθ1TheBayesfactoristheratiooftheposterioroddstotheprioroddsofthehypothesesposterioroddsp0/p1BF==.prioroddsπ0/π1ThestatisticBFisameasureoftheevid

13、enceprovidedbythedatainsupportofthehypothesisH0.TheposteriorprobabilityofthehypothesisH0canbeexpressedasafunctionoftheBayesfactorandthepriorprobabilitiesofthehypothesesbyπ0BFp0=.π0BF+1−π08.3AOne-SidedTestofaNormalMeanInanexamplefromChapter14ofBerry(1996),theauthorwasinterestedindet

14、ermininghistrueweightfroma