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1、FreeEnergySequentialMonteCarlo,ApplicationtoMixtureModelling*UniversityPressScholarshipOnlineOxfordScholarshipOnlineBayesianStatistics9JoséM.Bernardo,M.J.Bayarri,JamesO.Berger,A.P.Dawid,DavidHeckerman,AdrianF.M.Smith,andMikeWestPrintpublicationdate:2011PrintISBN-13:9780199694587PublishedtoOxfordSch
2、olarshipOnline:January2012DOI:10.1093/acprof:oso/9780199694587.001.0001FreeEnergySequentialMonteCarlo,ApplicationtoMixtureModelling*NicolasChopinPierreJacobDOI:10.1093/acprof:oso/9780199694587.003.0003AbstractandKeywordsWeintroduceanewclassofSequentialMonteCarlo(SMC)methods,whichwecallfreeenergySMC
3、.Thisclassisinspiredbyfreeenergymethods,whichoriginatefromphysics,andwhereonesamplesfromabiaseddistributionsuchthatagivenfunctionξ(θ)ofthestateθisforcedtobeuniformlydistributedoveragiveninterval.Fromaninitialsequenceofdistributions(πt)ofinterest,andaparticularchoiceofξ(θ),afreeenergySMCsamplercompu
4、tessequentiallyasequenceofbiaseddistributions(π̃t)withthefollowingproperties:(a)themarginaldistributionofξ(θ)withrespecttoπ̃tisapproximativelyuniformoveraspecifiedinterval,and(b)π̃tandπthavethesameconditionaldistributionwithrespecttoξ.Weapplyourmethodologytomixtureposteriordistributions,whicharehig
5、hlymultimodal.Inthemixturecontext,forcingcertainhyper‐parameterstohighervaluesgreatlyfacilitatesmodeswapping,andmakesitpossibletoPage1of31FreeEnergySequentialMonteCarlo,ApplicationtoMixtureModelling*recoverasymmetricoutput.WeillustrateourapproachwithunivariateandbivariateGaussianmixturesandtworeal‐
6、worlddatasets.Keywords:Freeenergybiasing,Labelswitching,Mixture,SequentialMonteCarlo,particlefilterSummaryWeintroduceanewclassofSequentialMonteCarlo(SMC)methods,whichwecallfreeenergySMC.Thisclassisinspiredbyfreeenergymethods,whichoriginatefromphysics,andwhereonesamplesfromabiaseddistributionsuchtha
7、tagivenfunctionξ(θ)ofthestateθisforcedtobeuniformlydistributedoveragiveninterval.Fromaninitialsequenceofdistributions(πt)ofinterest,andaparticularchoiceofξ(θ),afreeenergySMCsamplercomputessequentiallyaseque
