PUB IRMA LILLE Vol No IV

-

English
22 Pages
Read an excerpt
Gain access to the library to view online
Learn more

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


Subjects

Informations

Published by
Reads 66
Language English
Document size 3 MB
Report a problem
PUB.IRMoA,LILLE2007Vol.67,NIVAversatileMCMCstrategyforsamplingposteriordistributionsofanalyticallyintractablemodelsR.S.StoicaUniversite´Lille1LaboratoirePaulPainleve´59655Villeneuved’AscqCedexFranceP.GregoriandJ.MateuUniversitatJaumeICampusRiuSec,E-12071Castellon,SpainAbstractThispaperproposesanewversatilestrategyforsamplingposteriordistributionsforanalyticallyintractablemodels.BuildingsuchsamplersusingMarkovChainMonteCarlomethodologyusuallyleadstoalgorithmswhichareratherexpensivefromacomputationalpointofview,hencehav-ingveryfewchancestobeusedforpracticalapplications.Thestrategyweproposeovercomesthisdrawbackandiseasytouse.Adirectapplica-tionoftheproposedmethodisshownanddiscussed:maximumlikelihoodestimationoftheparametersofaspatialpointprocess.R´esume´Cetarticleproposeuneme´thodologiesouplepourlasimulationdesloisaposterioridesmode`lesayantdesconstantesdenormalisationquinesontpascalculablesanalytiquement.Utilisernaı¨vementlaphilosophieMonteCarlopourcetypedeproble`me,ame`nea`desalgorithmescouˆteuxdupointdevuedutempsdecalcul,doncpeuutilisablesenpratique.Lastrate´giequenousproposonse´liminecettedifficulte´etdepluselleestfacilea`utiliser.Nousmontronsetdiscutonsuneapplicationpratique:estimationdesparame`tresd’unprocessusponctuelparlame´thodedumaximumdevraisemblance.MathematicsSubjectClassifications(2000):60J22,60G55Keywords:computationalmethodsinMarkovchains,maximumlikelihoodestimation,pointprocesses.Note:partoftheworkofthefirstauthorwasdoneduringhisstaysatUniversityJaumeIandINRAAvignon.
I3V1Generalcontextandpresentationoftheprob-melThemodernprobabilitytheorytogetherwiththeincreasingperformancesofthecomputersallownowadaystheuseofcomplexstochasticmodelsinnumerousapplicationdomainssuchasenvironmentalsciences,astronomyorimageanaly-sis.Thedynamicsofadisappearingspecies,thedistributionofgalaxiesinouruniverse,orbrainactivityimagescanbeseenastherealisationofstochasticprocessescontrolledbysomeparameters.Scientistswishtoanswertwomajorquestions.Thefirstoneishowthemodelbehavesknowingthatitisgovernedbysomegivenparameters.Thesecondoneisthereverseofthefirstquestion.Thatis,whataretheparametersofthemodelthatcanmimicanobservedphenomenon.Inprobabilisticterms,answerstothesecondquestioncanbeformulatedifonecansamplefromtheposteriorlaw,i.e.thelawofthemodelparameters,conditionedtotheobservationofthephenomenonofinterest.Thereisalwaysabalancebetweenthemodelcapacityto“catch”realityanditsmathematicalproperties.Usually,realisticmodelsarerather“complicated”,hencetheirmathematicalpropertiesmaylook“notsoappealing”.Oneofthemostcommondrawbackexhibitedbysuchmodelsisthattheyareoftenana-lyticallyintractable.Therefore,computationsrelatedtosuchmodelscannotbedoneusingexactmathematicalformulae.Thesolutiontothatproblemistoperformthesecomputationsbymeansofcomputersimulations.Thesearereasonswhytodaythereexistsachallengeindevelopingstatisticalmethodologyallowinginferenceforsuchcomplexmodels.Theproposedsolu-tionshavetomeetseveralrequirementssuchas:mathematicalrigour,simplicityandcomputationalefficiency.Thisisthecontextwithinourpaperissituated.Inthefollowing,theproblemweaddressisintroduced:buildingaversatilesamplerforposteriordistributionsofanalyticallyintractablemodels.LetνbetheLebesguemeasureinRd,ΘacompactsubsetofRdofpositivemeasure,and(Θ,TΘ)thenaturalrestrictionofthemeasurespace(Rd,T).LetusconsiderastochasticprocessY,thatisdefinedonaprobabilityspace,F,)andcharacterisedbytheprobabilitydensityp(y|θ)withrespecttothereferencemeasure.θΘrepresentsthemodelparametervector.Fortheprobabilitydensitywewritep(y|θ)=f(y;θ))θ(cwithf(y;θ):Ω×ΘR+andc(θ)thenormalisingconstantgivenbyZc(θ)=f(y;θ)d(y).ΩThemodelp(y|θ)issaidtobeanalyticallyintractableifananalyticalexpres-sionisavailableforf(y;θ)butnotforc(θ).Theusualconditionsthatf(y;θ)has