Gibbs sampling for fitting finite and infinite gaussian mixture models

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1、GibbssamplingforttingniteandinniteGaussianmixturemodelsHermanKamperh.kamper@sms.ed.ac.uk14November20131IntroductionThisdocumentgivesahigh-levelsummaryofthenecessarydetailsforimplementingcollapsedGibbssamplingforttingGaussianmixturemodels(GMMs)followingaBayes

2、ianapproach.Thedocumentstructureisasfollows.Afternotationandreferencesections(Sections2and3),thecaseforsamplingtheparametersofaniteGaussianmixturemodelisdescribedinSection4.ThisisthenextendedtotheinnitecaseinSection5.Muchofthisdocumentisbasedoncontentfrom[1].I

3、recommendreadingthedocumentinconjunctionwithSections24.2and25.2in[1]whileconsultingtheotherreferencesgiventhroughoutthistext.2NotationWeaimtofollowgenerallythesamenotationasthatusedin[1].Belowisa(limited)summaryofthenotationused.2.1DataNNumberofdatavectors.DDime

4、nsionofdatavectors.xi2RDTheithdatavector.X=fx1;x2;:::;xNgSetofdatavectors.XniAlldatavectorsapartfromxi.XkSetofdatavectorsfrommixturecomponentk.XkniSetofdatavectorsfrommixturecomponentk,withouttakingxiintoaccount.NkNumberofdatavectorsfrommixturecomponentk.NkniNum

5、berofdatavectorsfrommixturecomponentk,withouttakingxiintoaccount.12.2ModelparametersKNumberofcomponentsinanitemixturemodel.zi2f1;2;:::;KgDiscretelatentstateindicatingwhichcomponentobservationxibelongsto.z=(z1;z2;:::;zN)Latentstatesforallobservationsx1;x2;:::;xN

6、.zniAlllatentstatesexcludingzi.MeanvectorofamultivariateGaussiandensity.Asubscriptisusedtoforaparticularcomponentinamixturemodel,e.g.k.CovariancematrixofamultivariateGaussiandensity.Asubscriptisusedforaparticularcomponentinamixturemodel,e.g.k.k=P(zi=k)Prior

7、probabilitythatdatavectorxiwillbeassignedtomixturecomponentk.=(1;2;:::;K)PriorassignmentprobabilityforallKcomponents.2.3Hyper-parameters=(1;2;:::;K)ParameterforDirichletprioronthemixingweights.=(m0;0;0;S0)ParametersfortheGaussian-inverse-Wishartprioronm

8、eanvectorandcovariancematrixofamultivariateGaussiandistribution.Theinterpretationfortheindividualparametersaregivenbelow.m0Priormeanfor.0Howstronglywebelievetheab