HammersleyClifford定理.docx

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