Markov Chain Monte Carlo and Gibbs Sampling

《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