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1、6MarkovChainMonteCarloMethods6.1IntroductionInChapter5,weintroducedtheuseofsimulationinBayesianinference.Rejec-tionsamplingisageneralmethodforsimulatingfromanarbitraryposteriordistribution,butitcanbedi?culttosetupsinceitrequirestheconstruc-tionofasuita
2、bleproposaldensity.ImportancesamplingandSIRalgorithmsarealsogeneral-purposealgorithms,buttheyalsorequireproposaldensitiesthatmaybedi?cultto?ndforhigh-dimensionalproblems.Inthischapter,weillustratetheuseofMarkovchainMonteCarlo(MCMC)algorithmsinsummarizi
3、ngposteriordistributions.Markovchainsareintroducedinthedis-cretestatespacesituationinSection6.2.Throughasimplerandomwalkexample,weillustratesomeoftheimportantpropertiesofaspecialMarkovchain,andweuseRtosimulatefromthechainandmovetowardthesta-tionarydist
4、ribution.InSection6.3,wedescribetwovariantsofthepopularMetropolis-HastingsalgorithmsinsettingupMarkovchains,andinSection6.4wedescribeGibbssampling,wheretheMarkovchainissetupthroughtheconditionaldistributionsoftheposterior.Wedescribeonestrategyforsum-ma
5、rizingaposteriordistributionandillustrateitforthreeproblems.MCMCalgorithmsareveryattractiveinthattheyareeasytosetupandprogramandrequirerelativelylittlepriorinputfromtheuser.Risaconvenientlanguageforprogrammingthesealgorithmsandisalsoverysuitableforperf
6、ormingout-putanalysis,whereonedoesseveralgraphicalandnumericalcomputationstocheckifthealgorithmisindeedproducingdrawsfromthetargetposteriordistribution.6.2IntroductiontoDiscreteMarkovChainsSupposeapersontakesarandomwalkonanumberlineonthevalues1,2,3,4,5
7、,6.Ifthepersoniscurrentlyataninteriorvalue(2,3,4,or5),inthenextsecondsheisequallylikelytoremainatthatnumberormovetoanadjacentJ.Albert,BayesianComputationwithR,UseR,DOI10.1007/978-0-387-92298-06,?SpringerScience+BusinessMedia,LLC20091186MarkovChainMonte
8、CarloMethodsnumber.Ifshedoesmove,sheisequallylikelytomoveleftorright.Ifthepersoniscurrentlyatoneoftheendvalues(1or6),inthenextsecondsheisequallylikelytostaystillormovetotheadjacentlocation.ThisisasimpleexampleofadiscreteMarkovchain.AMar
