RJMCMC

1、ReversiblejumpMCMC•Mixtureproblems•Variableselectioninregression•DetectingmultiplechangepointsRJMCMC(cont)RJMCMC(cont)RJMCMC(cont)•MovebetweendifferentparameterspacesRJMCMC•Algorithm–Acceptproposedvaluewithprobability–Otherwise,retainexistingvalues•DimensionmatchingRJMCMC(cont)UseinversetransformationTheacceptanceprobabilityisAnexample:finitemixtureestimation•RichardsonandGreen(1997)•Univariatenormalmixture•AnalternativerepresentationFinitemixtureestimation(cont)•Priorspecification–Independent–。

2、Foridentifiability–Weights–NumberofcomponentsAgraphicalrepresentationFinitemixtureestimation(cont)•Jointdistribution•AlgorithmDifferenttypeofmovesCombinemoveSplitmoveSplitmove(cont)Birth/deathmoveCalculationoftheacceptanceprobabilityPosteriorAchange-pointexample•Coalminingdisasterdata–Datesof192disastersfrom1851-1962–Model:aninhomogeneousPoissonprocesswithratex(t)–Thelog-likelihoodis•Modelx(t)asapiecewiseconstantfunction(stepfunction)withanunknownnumberofchange-pointsAchange-pointexample(cont)•P。

3、rior–Thenumberofchange-points:k~Poisson(λ)–Steppositions:s1,…,sk•Distributedastheeven-numberedorderstatisticsfrom2k+1pointsuniformlyon[1851,1962]•Ifusingk+1points,therecouldbetoomanyshortintervalswithnodata.–Heightsofsteps:iidGammaAnchange-pointexample(cont)•Possiblemoves–Birthmove(addstep)–Deathmove(deletestep)–Changearandomheight•Proposesuchthatisuniformon[-1/2,1/2]–Changearandomstep•Proposeuniformon(sj-1,sj+1)Achange-pointexample(cont)•Birthmove–Chooseapositionforthenewstepuniformlyfromtheint。

4、erval–Chooseaheightforthetwostepsbordering,jandj+1•Theoldheightshouldbeacompromisebetweenthetwonewheights•Useaweightedgeometricmean•Meanwhile,letu~uniform(0,1)•Increasethedimensionby2:newheight,newsteplocationAchange-pointproblem(cont)•Deathmove–Selectastepboundarysj+1atrandomtoremove–Newheightisdefinedby。