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1、GraphicalModelsJinlongWu&TiejunLiNov20081IntroductionJordan[3]presentesaveryconciseintroductiontoGraphicalModels(GMs):Graphicalmodelsareamarriagebetweenprobabilitytheoryandgraphtheory.Theyprovideanaturaltoolfordealingwithtwoproblemsthatoccurthroughoutapplied
2、mathematicsandengineering—uncer-taintyandcomplexity—andinparticulartheyareplayinganincreasinglyimportantroleinthedesignandanalysisofmachinelearningalgorithms.Fundamentaltotheideaofagraphicalmodelisthenotionofmodularity—acomplexsystemisbuiltbycombiningsimpler
3、parts.Probabilitytheoryprovidesthegluewherebythepartsarecombined,ensuringthatthesystemasawholeisconsistent,andprovidingwaystointerfacemodelstodata.Thegraphtheoreticsideofgraphicalmodelsprovidesbothanintuitivelyappeal-inginterfacebywhichhumanscanmodelhighly-i
4、nteractingsetsofvariablesaswellasadatastructurethatlendsitselfnaturallytothedesignofef?cientgeneral-purposealgorithms.Manyoftheclassicalmultivariateprobabilisticsystemsstudiedin?eldssuchasstatistics,systemsengineering,informationtheory,patternrecognitionands
5、tatisticalmechanicsarespecialcasesofthegeneralgraphicalmodelformalism—examplesincludemixturemodels,factoranalysis,hiddenMarkovmodels,Kalman?ltersandIsingmodels.Thegraphicalmodelframeworkprovidesawaytoviewallofthesesystemsasinstancesofacommonunderlyingformali
6、sm.Thisviewhasmanyadvantages—inparticular,specializedtechniquesthathavebeendevelopedinone?eldcanbetransferredbetweenresearchcommunitiesandexploitedmorewidely.Moreover,thegraphicalmodelformalismprovidesanaturalframeworkforthedesignofnewsystems.GMsareusuallydi
7、videdintotwotypes—undirectedanddirected.UndirectedGMsarealsocalledMarkovNetworksorMarkovRandomFields(MRFs),anddirectedGMsarealsoknownasBayesianNetworks(BNs),beliefnetworks,generativemodelsorcausalmodels.1.1DirectedGMs(BayesianNetworks)[4]InBayesianNetworks(B
8、Ns)eachvertexrepresentsarandomvariable,andanarcfromvertexXtovertexY(wealsosaidthatXisoneoftheparentsofY)meansXisoneofthereasonswhyYhappens,i.e.,XcausesY.HenceBNsareacyclic.BNsassumethatavariable
