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1、MarkovchainMonteCarloinaction:atutorialPeterJ.GreenUniversityofBristol,DepartmentofMathematics,Bristol,BS81TW,UK.P.J.Green@bristol.ac.uk1.IntroductionMarkovchainMonteCarloisprobablyabout50yearsold,andhasbeenbothdevelopedandextensivelyusedinphysicsforthelastfourdecades.However,themostspectacularinc
2、reaseinitsimpactandin
uenceinstatisticsandprobabilityhascomesincethelate'80's.Ithasnowcometobeanall-pervadingtechniqueinstatisticalcomputation,inparticularforBayesianinference,andespeciallyincomplexstochasticsystems.2.Cyclonesexample:pointprocessesandchangepointsWewillillustratetheideasofMCMCwitha
3、runningexample:theobservationsareapointprocessofeventsattimesy;y;:::;yinaninterval[0;L).Wesupposetheeventsoccurasa12NPoissonprocess
4、butatapossiblynon-uniformrate:sayx(t)perunittime,attimet;wewishtomakeinferenceaboutx(t).Weconsideraseriesofmodels,ultimatelyallowinganunknownnumberofchangepoints,unkn
5、ownhyperparameters,andaparametricperiodiccomponent.ThemodelsandtherespectivealgorithmsandinferenceswillbeillustratedbyananalysisofadatasetofthetimesofcycloneshittingtheBayofBengal;therewere141cyclonesoveraperiodof100years.Model1:constantrate.Firstsupposethatx(t)xforallt.Thenthetimesoftheeventsare
6、immaterial:weobserveNeventsinatimeintervaloflengthL;theobviousestimateofxNbisx=,themaximumlikelihoodestimatorofxundertheassumptionthatNhasaPoissonLdistribution,withmeanxL.Model2:constantrate,theBayesianway.ForaBayesianapproachtothisexample,supposethatwehavepriorinformationaboutx(frompreviousstudie
7、s,forexample).Supposewecanmodelthisbyx,( ;):Thenwe ndthataposteriorixhasaGammadistributionwithmean(+N)=(+L),orapproximatelyN=LifNandLarelargecomparedwithand.Thuswithalotofdata,theBayesianposteriormeanisclosetothemaximumlikelihoodestimator.ThereisnoneedforMCMCinthismodel:youcancalculatetheposterio
8、rexactly,andrecogniseitasastandarddistribution;itonlyworkedlikethisbecauseweusedaconjugateprior.Model3:constantrate,withhyperparameter.Supposeyouarereluctanttospecifyyourpriorfully:youarehappytosayx,( ;)andcansp
