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ID:40720831
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页数:24页
时间:2019-08-06
《Markov Chain Monte Carlo and Gibbs Sampling》由会员上传分享,免费在线阅读,更多相关内容在学术论文-天天文库。
1、MarkovChainMonteCarloandGibbsSamplingLectureNotesforEEB581,version26April2004°cB.Walsh2004AmajorlimitationtowardsmorewidespreadimplementationofBayesianap-proachesisthatobtainingtheposteriordistributionoftenrequirestheintegrationofhigh-dimensionalfunctions.Thiscanbecomputatio
2、nallyverydifficult,butseveralapproachesshortofdirectintegrationhavebeenproposed(reviewedbySmith1991,EvansandSwartz1995,Tanner1996).WefocushereonMarkovChainMonteCarlo(MCMC)methods,whichattempttosimulatedirectdrawsfromsomecomplexdistributionofinterest.MCMCapproachesareso-namedb
3、e-causeoneusestheprevioussamplevaluestorandomlygeneratethenextsamplevalue,generatingaMarkovchain(asthetransitionprobabilitiesbetweensamplevaluesareonlyafunctionofthemostrecentsamplevalue).Therealizationintheearly1990’s(GelfandandSmith1990)thatoneparticu-larMCMCmethod,theGibb
4、ssampler,isverywidelyapplicabletoabroadclassofBayesianproblemshassparkedamajorincreaseintheapplicationofBayesiananalysis,andthisinterestislikelytocontinueexpandingforsometimetocome.MCMCmethodshavetheirrootsintheMetropolisalgorithm(MetropolisandUlam1949,Metropolisetal.1953),a
5、nattemptbyphysiciststocomputecom-plexintegralsbyexpressingthemasexpectationsforsomedistributionandthenestimatethisexpectationbydrawingsamplesfromthatdistribution.TheGibbssampler(GemanandGeman1984)hasitsoriginsinimageprocessing.Itisthussomewhatironicthatthepowerfulmachineryof
6、MCMCmethodshadessentiallynoimpactonthefieldofstatisticsuntilratherrecently.Excellent(anddetailed)treatmentsofMCMCmethodsarefoundinTanner(1996)andChaptertwoofDraper(2000).Additionalreferencesaregivenintheparticularsectionsbelow.MONTECARLOINTEGRATIONTheoriginalMonteCarloapproac
7、hwasamethoddevelopedbyphysiciststouserandomnumbergenerationtocomputeintegrals.SupposewewishtocomputeacomplexintegralZbh(x)dx(1a)aIfwecandecomposeh(x)intotheproductionofafunctionf(x)andaprobability12MCMCANDGIBBSSAMPLINGdensityfunctionp(x)definedovertheinterval(a;b),thennotetha
8、tZbZbh(x)dx=f(x)p(x)dx=Ep(x)[f(x)](1b)aasothattheintegralcanbeexpressedasan
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