Using Markov chain Monte Carlo methods for estimat

arXiv:gr-qc/0102018v15Feb2001UsingMarkovchainMonteCarlomethodsforestimatingparameterswithgravitationalradiationdata.NelsonChristensen1andRenateMeyer2∗1PhysicsandAstronomy,CarletonCollege,Northfield,Minnesota,55057,USA2DepartmentofStatistics,TheUniversityofAuckland,Auckland,NewZealand(February3,2008)AbstractWepresentaBayesianapproachtotheproblemofdeterminingparametersforcoalescingbinarysystemsobservedwithlaserinterferometricdetectors.ByapplyingaMarkovChainMonteCarlo(MCMC)algorithm,specificallytheGibbssampler,wedemonstratethepotentialthatMCMCtechniquesmayholdforthecomputationofposteriordistributionsofparametersofthebinarysystemthatcreatedthegravityradiationsignal.WedescribetheuseoftheGibbssamplermethod,andpresentexampleswherebysignalsaredetectedandanalyzedfromwithinnoisydata.04.80.Nn,02.70.Lq,06.20.DqTypesetusingREVTEX1I.INTRODUCTIONAnumberofcollaborationsaroundtheworldwillbeoperatinglaserinterferometricgravitationradiationantennaswithinthenextfewyears.IntheUnitedStatestheLaserInterferometricGravitationalWaveObservatory(LIGO)issoontobeoperational,with4kmarmlengthinterferometersinHanford,Washington,andLivingston,Louisiana[1].AsimilarFrench-ItaliandetectorwillbebuiltinEurope(VIRGO)[2,3].Coalescingbinariescontainingneutronstars(NS)orblackholes(BH)arelikelytobethecleanestandmostpromisingsourceofdetectableradiation[4].UltimatelytheLIGO-VIRGOnetworkmayobservebinariesouttoadistanceof2Gpc[5].Thedetectionofcoalescingbinaryeventswillprovidephysicistswithextremelyusefulcosmologicalinformation.Ini-tiallySchutz[6]notedthatadetectedsignalcontainsenoughinformationtodeciphertheabsolutedistancetothesystem,andhencethedeterminationoftheHubbleconstantwouldbeachievedthroughtheobserveddistributionofseveralbinaries.Subsequentwork[7]indi-catesthattheuncertaintyinthemeasureddistancecanbecomparabletothedistanceitself,butimportantcosmologicaltestswillstillbepossiblethroughtheobservationofnumerousmergers[8].Inadditiontothecosmologicalimportance,accurateparameterestimationintheob-servedcoalescingbinarieswillprovideahostofinformationofgreatphysicalsignificance.ObservationofthetimeoftidaldisruptionofanNS-NSbinarysystemmaypermitadeter-minationoftheNSradiiandinformationontheNSequationofstate[9].Thecharacteristicsofradiationinthepost-Newtonianregimewillprovideinsightintohighlynon-lineargeneralrelativisticeffects[7,10,11].TheformationofaBHattheendofaNS-NScoalescence,orthemergeroftwoBHs,willproducegravitationalradiationasthesystemdecaystoaKerrBH;thisisanextremelyinterestingradiationproductionregime[10,11].ApplicationofBayes’theoremiswellsuitedtoastrophysicalobservations[12].TheBayesianversusfrequentistapproachestogravitationalradiationdataanalysisarewellpre-sentedby[13].Parameterestimationfromthegravitywavesignalsofcoalescingcompact2binariesprovidesanimportantapplicationofBayesianmethods[5,7,14,15].DifficultieswiththecalculationofBayesianposteriordistributionshavebeenovercomebytherapiddevel-opmentofMarkovChainMonteCarlo(MCMC)methodsinthelastdecade(see[16]foranintroduction).AlthoughtheinitialMCMCalgorithmdatesbackto[17],theenormouspotentialthatMCMCmethodsmightholdforBayesianposteriorcomputationsremainedlargelyunrecognizedwithinthestatisticalcommunityuntiltheseminalpaperbyGemanandGeman[18]inthecontextofdigitalimageanalysis.Sincethen,MCMCmethodshavehadahugeimpactonmanyareasofappliedstatistics.IthasnowbecomepracticaltoapplyBayesianmethodstocomplexproblems.Thus,weexpectasimilareffectongravitationalwavedataanalysis.Theinitialgoalofourresearcheffort,presentedinthispaper,istodemonstratetheusefulnessofMCMCtechniquesforestimatingparametersfromcoalescingbinarysignalsdetectedbylaserinterferometricantennas.TheGibbssampler[16]isoneofthesimplerMCMCtechniques,andweuseitasastartingpointforourinvestigationprimarilybecausethereisreadilyavailablesoftware[19].Ourstudyofgravitywavesignalsisconductedto2.5post-Newtonian(PN)order.Thesignalsdependonfiveindependentparameters;themassesofthetwocompactobjects,theamplitudeofthedetectedsignal,thecoalescencetimeandthephaseofthesignalatcoalescence.TheBayesiantechniquesweemploywillnotonlygivepointestimatesoftheseparameters,butalsoproducetheircompleteposteriorprobabilitydistributionthatcanbeemployedtosummarizetheuncertaintyofparameterestimatesthroughposteriorcredibilityintervals,forinstance.Incontrasttofrequentistconfidenceintervals,thesedonotrelyonlargesampleasymptoticsandhaveasimple,naturalinterpretation.Thepaperisorganizedasfollows:InSectionIIwebrieflyreviewBayesianinferenceanddescribetheMCMCsimulationtechniqueweuse,specificallytheGibbssamplerandsoftwareforitsimplementation.InSectionIIIwepresenttwoexampleswhereweuseourMCMCapproachtoidentifytheparameterswhichcreatedthesignalthatisburiedinsynthesizedLIGOnoise.InSectionIVweanalyzeanumberofissuesthatwilleffecttheefficiency3andcalculationaltimeofaMCMCapproachtothecoalescingbinaryparameterestimationproblem.WeconcludewithadiscussionofourresultsandthedirectionoffutureeffortsinSectionV.II.BAYESIANINFERENCEANDPOSTERIORCOMPUTATIONWebrieflyreviewtheBayesianapproachtoparameterestimation.Le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