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1、1IntroductiontoMarkovChainMonteCarloCharlesJ.Geyer1.1HistoryDespiteafewnotableusesofsimulationofrandomprocessesinthepre-computerera(HammersleyandHandscomb,1964,Section1.2;Stigler,2002,Chapter7),practicalwidespreaduseofsimulationhadtoawaittheinventionofcomputers.Almostassoonascomputerswere
2、invented,theywereusedforsimulation(HammersleyandHandscomb,1964,Section1.2).ThenameMonteCarlostartedascutenessgamblingwasthen(around1950)illegalinmostplaces,andthecasinoatMonteCarlowasthemostfamousintheworldbutitsoonbecameacolorlesstechnicaltermforsimulationofrandomprocesses.MarkovchainMon
3、teCarlo(MCMC)wasinventedsoonafterordinaryMonteCarloatLosAlamos,oneofthefewplaceswherecomputerswereavailableatthetime.Metropolisetal.(1953)∗simulatedaliquidinequilibriumwithitsgasphase.Theobviouswaytofindoutaboutthethermodynamicequilibriumistosimulatethedynamicsofthesystem,andletitrununtili
4、treachesequilibrium.Thetourdeforcewastheirrealizationthattheydidnotneedtosimulatetheexactdynamics;theyonlyneededtosimulatesomeMarkovchainhavingthesameequilibriumdistribution.SimulationsfollowingtheschemeofMetropolisetal.(1953)aresaidtousetheMetropolisalgorithm.Ascomputersbecamemorewidelya
5、vailable,theMetropolisalgorithmwaswidelyusedbychemistsandphysicists,butitdidnotbecomewidelyknownamongstatisticiansuntilafter1990.Hastings(1970)general-izedtheMetropolisalgorithm,andsimulationsfollowinghisschemearesaidtousetheMetropolisHastingsalgorithm.AspecialcaseoftheMetropolisHastingsa
6、lgorithmwasintroducedbyGemanandGeman(1984),apparentlywithoutknowledgeofearlierwork.SimulationsfollowingtheirschemearesaidtousetheGibbssampler.MuchofGemanandGeman(1984)discussesoptimizationtofindtheposteriormoderatherthansimulation,andittooksometimeforittobeunderstoodinthespatialstatisticsc
7、ommunitythattheGibbssamplersimulatedtheposteriordistribution,thusenablingfullBayesianinferenceofallkinds.AmethodologythatwaslaterseentobeverysimilartotheGibbssamplerwasintroducedbyTannerandWong(1987),againapparentlywithoutknowledgeofearlierwork.Tothisday,somereferto
