bayesian analysis large sample methods

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1、LargeSampleMethodsInordertomakeBayesianinferenceabouttheparameter0^givenamodelf{x6)^oneneedstochooseanappropriatepriordistributionfor6.Giventhedatax,thepriordistributionisusedtofindtheposteriordistributionandvar•iousposteriorsummarymeasures,dependingontheproblem.Thuse

2、xactorapproximatecomputationoftheposteriorisamajorproblemforaBayesian.Undercertainregularityconditions,theposteriorcanbeapproximatedbyanormaldistributionwiththemaximumlikelihoodestimate(MLE)asthemeanandinverseoftheobservedFisherinformationmatrixasthedispersionmatrix,if

3、thesamplesizeislarge.Ifmoreaccuracyisneeded,onemayusetheKass-Kadane-TierneyorEdgeworthtyperefinements.Alternatively,onemaysamplefromtheapproximateposteriorandtakeresorttoimportancesam•pling.Posteriornormalityhasanimportantphilosophicalimplication,whichwediscussbelow.Ho

4、wtheposteriorinferenceisinfluencedbyaparticularpriordependsontherelativemagnitudeoftheamountofinformationinthedata,whichfori.i.d.observationsmaybemeasuredbythesamplesizenornl{6)orobservedFisherinformation7^(definedinSection4.1.2),andtheamountofinforma•tionintheprior,wh

5、ichisdiscussedinChapter5.Asthesamplesizegrows,theinfluenceofthepriordistributiondiminishes.Thusforlargesamples,aprecisemathematicalspecificationofpriordistributionisnotnecessary.Inmostcasesoflow-dimensionalparameterspace,thesituationislikethis.ABayesianwouldrefertoitas

6、washingawayofthepriorbythedata.Thereareseveralmathematicalresultsembodyingthisphenomenonofwhichposteriornormalityisthemostwell-known.Thischapterdealswithposteriornormalityandsomeofitsrefinements.WebeginwithadiscussiononlimitingbehaviorofposteriordistributioninSection4.

7、1.Asketchofproofofasymptoticnormalityofposteriorisgiveninthissection.AmoreaccurateposteriorapproximationbasedonLaplace'sasymptoticmethodanditsrefinementsbyTierney,Kass,andKadanearethesubjectsofSection4.3.Arefinementofposteriornormalityisdiscussed1004LargeSampleMethodsi

8、nSection4.2whereanasymptoticexpansionoftheposteriordistributionwithaleadingnormaltermisoutlined.Throughoutthischapter