HammersleyClifford定理.docx
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HammersleyClifford定理.docx
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HammersleyClifford定理
HammersleyClifford定理
Hammersley–CliffordtheoremTheHammersley–Cliffordtheoremisaresultinprobabilitytheory,mathematicalstatisticsandstatisticalmechanics,thatgivesnecessaryandsufficientconditionsunderwhichapositiveprobabilitydistributioncanberepresentedasaMarkovnetwork(alsoknownasaMarkovrandomfield).ItstatesthataprobabilitydistributionthathasapositivemassordensitysatisfiesoneoftheMarkovpropertieswithrespecttoanundirectedgraphGifandonlyifitisaGibbsrandomfield,thatis,itsdensitycanbefactorizedoverthecliques(orcompletesubgraphs)ofthegraph.
TherelationshipbetweenMarkovandGibbsrandomfieldswasinitiatedbyRolandDobrushin[1]andFrankSpitzer[2]inthecontextofstatisticalmechanics.ThetheoremisnamedafterJohnHammersleyandPeterCliffordwhoprovedtheequivalenceinanunpublishedpaperin1971.[3][4]Simplerproofsusingtheinclusion-exclusionprincipleweregivenindependentlybyGeoffreyGrimmett,[5]Preston[6]andSherman[7]in1973,withafurtherproofbyJulianBesagin1974.[8]
Notes
^Dobrushin,P.L.(1968),"TheDescriptionofaRandomFieldbyMeansofConditionalProbabilitiesandConditionsofItsRegularity",TheoryofProbabilityanditsApplications13
(2):
197–224,doi:
10.1137/1113026,Spitzer,Frank(1971),"MarkovRandomFieldsandGibbsEnsembles",TheAmericanMathematicalMonthly78
(2):
142–154,doi:
10.2307/2317621,JSTOR2317621,Hammersley,J.M.;Clifford,P.(1971),Markovfieldsonfinitegraphsandlattices,Clifford,P.(1990),"Markovrandomfieldsinstatistics",inGrimmett,G.R.;Welsh,D.J.A.,DisorderinPhysicalSystems:
AVolumeinHonourofJohnM.Hammersley,OxfordUniversityPress,pp.19–32,ISBN0198532156,MR1064553,retrieved2009-05-04^Grimmett,G.R.(1973),"Atheoremaboutrandomfields",BulletinoftheLondonMathematicalSociety5
(1):
81–84,doi:
10.1112/blms/5.1.81,MR0329039^Preston,C.J.(1973),"GeneralizedGibbsstatesandMarkovrandomfields",AdvancesinAppliedProbability5
(2):
242–261,doi:
10.2307/1426035,JSTOR1426035,MR0405645.JSTOR1426035,Sherman,S.(1973),"MarkovrandomfieldsandGibbsrandomfields",IsraelJournalofMathematics14
(1):
92–103,doi:
10.1007/BF02761538,MR0321185^Besag,J.(1974),"Spatialinteractionandthestatisticalanalysisoflatticesystems",JournaloftheRoyalStatisticalSociety.SeriesB(Methodological)36
(2):
192–236,MR0373208.JSTOR2984812FurtherreadingBilmes,Jeff(Spring2006),Handout2:
Hammersley–Clifford,coursenotesfromUniversityofWashingtoncourse.Grimmett,Geoffrey,ProbabilityonGraphs,Chapter7,Helge,TheHammersley–CliffordTheoremanditsImpactonModernStatistics,probability-relatedarticleisastub.YoucanhelpWikipediabyexpandingit.
Retrievedfrom"–Clifford_theorem"FromWikipedia,thefreeencyclopediaThefirstafternoonofthememorialsessionforJulianBesaginBristolwasanintenseandattimesemotionalmoment,wherefriendsandcolleaguesofJuliansharedmemoriesandstories.Thiscollectionoftributesshowedhowmuchofalarger-than-lifecharacterhewas,fromhislong-termedandwide-rangedimpactonstatisticstohisveryhighexpectations,bothforhimselfandforothers,leadingtoatotalanduncompromisingresearchethics,tohispassionfor[extreme]sportsandoutdoors.(Thestoriesduringandafterdinerwereofamorepersonalnature,butatleastasmuchenjoyable!
)ThetalksontheseconddayshowedhowmuchandhowdeeplyJulianhadcontributedtospatialstatisticsandagriculturalexperiments,topseudo-likelihood,toMarkovrandomfieldsandimageanalysis,andtoMCMCmethodologyandpractice.IhopeIdidnotbotchtoomuchmypresentationonthehistoryofMCMC,whileIfoundreadingthroughthe1974,1986and1993ReadPapersandtheirdiscussionsanimmenselyrewardingexperiment(IwishIhaddonepriortocompletingourStatisticalSciencepaper,butitwasboundtobeincompletebynature!
).SomeinterestinglinksmadebytheaudiencewerethepriorpublicationofproofsoftheHammersley-Cliffordtheoremin1973(byGrimmet,Preston,andSteward,respectively),aswellastheproposalofaGibbssamplerbyBrianRipleyasearlyas1977(eventhoughHastingsdiduseGibbsstepsinoneofhisexamples).ChristopheAndrieualsopointedouttomeaveryearlyMonteCarloreviewbyJohnHaltoninthe1970SIAMRewiew,reviewthatIwillread(andcommment)assoonaspossible.Overall,IamquitegladIcouldtakepartinthismemorialandIamgratefultobothPetersfororganisingitasafittingtributetoJulian.MarkovChainMonteCarlo(MCMC)methodsarecurrentlyaveryactivefieldofresearch.MCMCmethodsaresamplingmethods,basedonMarkovChainswhichareergodicwithrespecttothetargetprobabilitymeasure.Theprincipleofadaptivemethodsistooptimizeontheflysomedesignparametersofthealgorithmwithrespecttoagivencriterionreflectingthesampler'sperformance(optimizetheacceptancerate,optimizeanimportancesampling
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