Objective Bayesian analysis of spatially correlate

ObjectiveBayesianAnalysisofSpatiallyCorrelatedDataJamesO.BERGER,VictorDEOLIVEIRA,andBrunoSANSO11JamesO.BergerisArtsandSciencesProfessor,InstituteofStatisticsandDecisionSciences,DukeUniversity,NC,U.S.A.,VictorDeOliveiraisAssociateProfessorandBrunoSansoisProfes-sor,DepartamentodeComputoCientcoyEstadstica,UniversidadSimonBolvar,Aptdo.89000,Caracas1080-A,Venezuela.Thisworkwassupported,inpart,bytheNationalScienceFounda-tion(USA),GrantsDMS-9802261andDMS-0103265,andbyConsejoNacionaldeInvestigacionesCientcasyTecnologicas(Venezuela),GrantG-97000592.TheauthorsaregratefultoMontserratFuentesandarefereeforhelpfulcommentsthatconsiderablyimprovedthepaper.AbstractSpatiallyvaryingphenomenaareoftenmodeledusingGaussianrandomelds,speciedbytheirmeanfunctionandcovariancefunction.Thespatialcorrelationstructureofthesemodelsiscommonlyspeciedtobeofacertainform(e.g.,spherical,powerexponential,rationalquadratic,orMatern)withasmallnumberofunknownparameters.WeconsiderobjectiveBayesiananalysisofsuchspatialmodels,whenthemeanfunctionoftheGaussianrandomeldisspeciedasinalinearmodel.Itisthusnecessarytodetermineanobjective(ordefault)priordistributionfortheunknownmeanandcovarianceparametersoftherandomeld.Werstshowthatcommonchoicesofdefaultpriordistributions,suchastheconstantpriorandtheindependentJereysprior,typicallyresultinimproperposteriordistributionsforthismodel.Next,thereferencepriorforthemodelisdeveloped,andisshowntoyieldaproperposteriordistribution.AfurtherattractivepropertyofthereferenceprioristhatitcanbeuseddirectlyforcomputationofBayesfactorsorposteriorprobabilitiesofhypothesestocomparedierentcorrelationfunctions,eventhoughthereferencepriorisimproper.Anillustrationisgivenusingaspatialdatasetoftopographicelevations.Keywords:Correlationfunction;Covarianceselection;Frequentistcoverage;Gaussianran-domeld;Geostatistics:Improperposterior;Integratedlikelihood;Jereysprior;Referenceprior.1IntroductionRandomeldsarepowerfulprobabilistictoolsformodelingspatialdataandmakinginferenceaboutspatiallyvaryingphenomena(Cressie,1993).TheBayesianapproachtoanalysisofspatialdatahasseenanupsurgeinpopularity,especiallywhenthegoalisprediction(e.g.DeOliveira,KedemandShort,1997;EckerandGelfand,1997;HandcockandStein,1993).ThemainadvantageoftheBayesianapproachisthatparameteruncertaintyisfullyaccountedforwhenperformingpredictionandinference,eveninsmallsamples.WeconsiderobjectiveBayesiananalysisofspatialdata,utilizingnon-informativeorcon-ventionalpriordistributionsforunknownparametersofGaussianrandomelds.Suchpriorsareoftenusedinspatialmodels,inpartbecauseofthedicultyofinterpreting(andhence,insubjectiveelicitationof)thecorrelationparameters.Themainmotivationforthispaperisthatcommonlyusednon-informativepriorscanfailtoyieldproperposteriordistribu-tions.ThiswasalludedtoinDeOliveiraetal.(1997)andstated,forparticularmodels,inStein(1999),butnosystematicstudyofthephenomenonhadbeenundertaken.Theprimarymethodologicalgoalofthispaperistoprovidenon-informativepriordistributionsforspatialmodelsthatdoresultinproperposteriordistributions,andthathaveadditionaldesirableproperties.Oneofthemostinterestingoftheseadditionalpropertiesisthattherecommendednon-informativepriorscanbedirectlyusedforcomputationofBayesfactorstocomparedierentpossiblespatialcovariancefunctions.Thechoiceoftheformofthespatialcovariancefunc-tionisoftenquitearbitrary,sothatitisparticularlyimportanttohavepowerfulandeasymethodsofcomparison.StandardBayesianmodelcomparisonisusuallynotpossiblewithnon-informativepriordistributions,sothatonemustresorttomoreinvolvedBayesiantech-nologies(e.g.,intrinsicBayesfactors,BergerandPericchi,1996orfractionalBayesfactors,O’Hagan,1995)whichcanbequitecomputationallyintensive.Hence,thefactthatthenon-informativepriorsdevelopedhereincanbedirectlyusedformodelcomparisoninspatialmodelsisaverywelcomesimplication.Thedevelopmentsinthepaperarealsoofinterestfromseveralfoundationalandtheoret-icalperspectives.First,thedicultywithposteriorproprietyforcommonnon-informativepriorsiscausedbythefactthat,whentheGaussianrandomeldhasanunknownmeanlevel,theintegratedlikelihoodofthecorrelationparametersistypicallyboundedawayfromzero;suchintegratedlikelihoodsareveryunusualinstatistics.Thesecondfeatureoffoundationalinterestisthattheoftenprescribed(independence)Jereysand(asymptotic)referencepriorscanfailtoyieldproperposteriors.Thisissurprising,inthatfewsuchexamplesareknownforthesetwopopularnon-informativepriors.Thespatialproblemthusjoinstheranksof‘basic’examplesthatindicatethelimitsofnon-informativepriortheories.Oureventuallyrecommendednon-informativepriorisan‘exact’referenceprior,devel-opedbyusingexactmarginalizationinthereferenceprioralgorithm,insteadofthemoretypicalasymptoticmarginalizationofBergerandBernardo(1992).Thisistherstknowninstanceinwhichexactmarginalization(asopposedtoasymptoticmarginalization)inthealgorithmisrequired,providinganotherinterestingtheoreticalmotivationforthestudy.Fi-nally,thephenomenonuncoveredhereappearstoalsoapplytonumerousnon-spatialmodels,includingmany