PUB IRMA LILLE Vol No IV

Paul Painleve

Description

PUB. IRMA, LILLE 2007 Vol. 67, No IV A versatile MCMC strategy for sampling posterior distributions of analytically intractable models R. S. Stoica? Universite Lille 1 Laboratoire Paul Painleve 59655 Villeneuve d'Ascq Cedex France P. Gregori†and J. Mateu‡ Universitat Jaume I Campus Riu Sec, E-12071 Castellon, Spain Abstract This paper proposes a new versatile strategy for sampling posterior distributions for analytically intractable models. Building such samplers using Markov Chain Monte Carlo methodology usually leads to algorithms which are rather expensive from a computational point of view, hence hav- ing very few chances to be used for practical applications. The strategy we propose overcomes this drawback and is easy to use. A direct applica- tion of the proposed method is shown and discussed: maximum likelihood estimation of the parameters of a spatial point process. Resume Cet article propose une methodologie souple pour la simulation des lois a posteriori des modeles ayant des constantes de normalisation qui ne sont pas calculables analytiquement. Utiliser naıvement la philosophie Monte Carlo pour ce type de probleme, amene a des algorithmes couteux du point de vue du temps de calcul, donc peu utilisables en pratique. La strategie que nous proposons elimine cette difficulte et de plus elle est facile a utiliser. Nous montrons et discutons une application pratique : estimation des parametres d'un processus ponctuel par la methode du maximum de vraisemblance.

• method computationally heavy


• philosophie monte


• density over


• variable method


• maximum likelihood


• processus ponctuel par la methode du maximum de vraisemblance


• models cannot


• carlo

Sujets

• Painlevé
• Maximum likelihood

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 66
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

PUB.IRM o A,LILLE2007 Vol.67,NIV AversatileMCMCstrategyforsampling posteriordistributionsofanalyticallyintractable models R.S.Stoica ∗ Universite´Lille1 LaboratoirePaulPainleve´ 59655Villeneuved’AscqCedexFrance P.Gregori † andJ.Mateu ‡ UniversitatJaumeI CampusRiuSec,E-12071Castellon,Spain Abstract Thispaperproposesanewversatilestrategyforsamplingposterior distributionsforanalyticallyintractablemodels.Buildingsuchsamplers usingMarkovChainMonteCarlomethodologyusuallyleadstoalgorithms whichareratherexpensivefromacomputationalpointofview,hencehav- ingveryfewchancestobeusedforpracticalapplications.Thestrategy weproposeovercomesthisdrawbackandiseasytouse.Adirectapplica- tionoftheproposedmethodisshownanddiscussed:maximumlikelihood estimationoftheparametersofaspatialpointprocess. R´esume´ Cetarticleproposeuneme´thodologiesouplepourlasimulationdes loisaposterioridesmode`lesayantdesconstantesdenormalisationqui nesontpascalculablesanalytiquement.Utilisernaı¨vementlaphilosophie MonteCarlopourcetypedeproble`me,ame`nea`desalgorithmescouˆteux dupointdevuedutempsdecalcul,doncpeuutilisablesenpratique.La strate´giequenousproposonse´liminecettedifficulte´etdepluselleest facilea`utiliser.Nousmontronsetdiscutonsuneapplicationpratique: estimationdesparame`tresd’unprocessusponctuelparlame´thodedu maximumdevraisemblance. MathematicsSubjectClassifications(2000):60J22,60G55 Keywords:computationalmethodsinMarkovchains,maximumlikelihood estimation,pointprocesses. Note:partoftheworkofthefirstauthorwasdoneduringhisstaysatUniversity JaumeIandINRAAvignon.

I3–V 1Generalcontextandpresentationoftheprob- mel Themodernprobabilitytheorytogetherwiththeincreasingperformancesofthe computersallownowadaystheuseofcomplexstochasticmodelsinnumerous applicationdomainssuchasenvironmentalsciences,astronomyorimageanaly- sis.Thedynamicsofadisappearingspecies,thedistributionofgalaxiesinour universe,orbrainactivityimagescanbeseenastherealisationofstochastic processescontrolledbysomeparameters.Scientistswishtoanswertwomajor questions.Thefirstoneishowthemodelbehavesknowingthatitisgoverned bysomegivenparameters.Thesecondoneisthereverseofthefirstquestion. Thatis,whataretheparametersofthemodelthatcanmimicanobserved phenomenon. Inprobabilisticterms,answerstothesecondquestioncanbeformulatedif onecansamplefromtheposteriorlaw, i.e. thelawofthemodelparameters, conditionedtotheobservationofthephenomenonofinterest. Thereisalwaysabalancebetweenthemodelcapacityto“catch”realityand itsmathematicalproperties.Usually,realisticmodelsarerather“complicated”, hencetheirmathematicalpropertiesmaylook“notsoappealing”.Oneofthe mostcommondrawbackexhibitedbysuchmodelsisthattheyareoftenana- lyticallyintractable.Therefore,computationsrelatedtosuchmodelscannotbe doneusingexactmathematicalformulae.Thesolutiontothatproblemisto performthesecomputationsbymeansofcomputersimulations. Thesearereasonswhytodaythereexistsachallengeindevelopingstatistical methodologyallowinginferenceforsuchcomplexmodels.Theproposedsolu- tionshavetomeetseveralrequirementssuchas:mathematicalrigour,simplicity andcomputationalefficiency. Thisisthecontextwithinourpaperissituated.Inthefollowing,theproblem weaddressisintroduced:buildingaversatilesamplerforposteriordistributions ofanalyticallyintractablemodels. LetνbetheLebesguemeasurein R d ,Θacompactsubsetof R d ofpositive measure,and(Θ,T Θ ,ν)thenaturalrestrictionofthemeasurespace( R d ,T,ν). Letusconsiderastochasticprocess Y ,thatisdefinedonaprobabilityspace (Ω,F,)andcharacterisedbytheprobabilitydensityp(y|θ)withrespectto thereferencemeasure.θ∈Θrepresentsthemodelparametervector.Forthe probabilitydensitywewrite p(y|θ)= f(y;θ) )θ(c withf(y;θ):Ω×Θ→ R + andc(θ)thenormalisingconstantgivenby Z c(θ)=f(y;θ)d(y). Ω Themodelp(y|θ)issaidtobeanalyticallyintractableifananalyticalexpres- sionisavailableforf(y;θ)butnotforc(θ).Theusualconditionsthatf(y;θ)has