Markov Chain Monte Carlo (MCMC)

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1、1MarkovChainMonteCarlo(MCMC)ByStevenF.ArnoldProfessorofStatistics-PennStateUniversitySomereferencesforMCMCare1.Tanner,M.(1993)ToolsforStatisticalInference,MethodforExplorationofPosteriorDistributionsandLikelihoodFunc-tions.2.Gilks,W.,Richardson,S.andSpiegelhalter,D.(1996)M

2、arkovChainMonteCarloinPractice.3.Gelman,A.,Carlin,J.,Stern,HandRubin,D.(1995)BayesianDataAnalysis.AreferenceforMarkovChainsis1.Ross,Sheldon,(1989)IntroductiontoProbabilitymodels4thEdit.11.1MCMCandBayesianStatisticsInthelast15yearstherehasbeenanexplosionofworkinBayesianstat

3、istics.AsyourecallaBayesianstatisticianchoosesapriordistributionovertheparameterspace.Hethendeterminestheposteriordistribution.AsDr.Leonardobserved,onceweknowtheposteriordistribution,Bayesiananalysisisoftenfairlyeasy.Oftenchoosingthepriorandcomputingtheposteriorarethehardp

4、arts.Inthepast,oneoftheproblemswithBayesianstatisticshasbeen¯ndingtheposteriordistribution.InrecentyearsthisproblemhasbeencontrolledbyusingMCMCtosimulatetheposterior.21.2MarkovchainsAdiscretetimeMarkovChainisasequenceofrandomvariablesinwhichtheconditionaldistributionofapre

5、sentobservationsgivenasetofpastobservationsonlydependsonthepastthroughthemostrecentobservation.Insymbols³´k1

6、Markovchainsaretime-homogenious.3Example(symmetricrandomwalk(drunkardswalk))ThisisaMarkovchainonthesetofallintegersinwhich(Xt¡1+1withp=:5XtjXt¡1=Xt¡1¡1withp=:5ThepossiblevaluesfortheMarkovchainarecalledthestatesoftheMarkovchain.Astationarydistribution¼foraMarkovchainisadis

7、tributionoverthestatessuchthatifwestarttheMarkovchainin¼,westayin¼.Alimitingdistribution¼;isadistributionoverthestatessuchthatwhateverthestartingthedistribution¼0,theMarkovchainconvergesto¼:Itiseasilyseenthatifthereisalimitingdistribution¼;thenitisunique,anditistheonlystat

8、ionarydistribution.Itiseasierto¯ndastationarydistributionthanalimitingdistribution.Soto¯n