An_introduction_to_graphical_models

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1、AnintroductiontographicalmodelsKevinP.Murphy10May20011IntroductionThefollowingquotation,fromthePrefaceof[Jor99],providesaveryconciseintroductiontographicalmodels.Graphicalmodelsareamarriagebetweenprobabilitytheoryandgraphtheory.Theyprovideanaturaltoolfordealingwithtwoproblemstha

2、toccurthroughoutappliedmathematicsandengineering{uncertaintyandcomplexity{andinparticulartheyareplayinganincreasinglyimportantroleinthedesignandanalysisofmachinelearningalgorithms.Fundamentaltotheideaofagraphicalmodelisthenotionofmodularity{acomplexsystemisbuiltbycombiningsimple

3、rparts.Probabilitytheoryprovidesthegluewherebythepartsarecombined,ensuringthatthesystemasawholeisconsistent,andprovidingwaystointerfacemodelstodata.Thegraphtheoreticsideofgraphicalmodelsprovidesbothanintuitivelyappealinginterfacebywhichhumanscanmodelhighly-interactingsetsofvaria

4、blesaswellasadatastructurethatlendsitselfnaturallytothedesignofecientgeneral-purposealgorithms.Manyoftheclassicalmultivariateprobabalisticsystemsstudiedineldssuchasstatistics,systemsengineering,informationtheory,patternrecognitionandstatisticalmechanicsarespecialcasesofthegene

5、ralgraphicalmodelformalism{examplesincludemixturemodels,factoranalysis,hiddenMarkovmodels,KalmanltersandIsingmodels.Thegraphicalmodelframeworkprovidesawaytoviewallofthesesystemsasinstancesofacommonunderlyingformalism.Thisviewhasmanyadvantages{inparticular,specializedtechniquest

6、hathavebeendevelopedinoneeldcanbetransferredbetweenresearchcommunitiesandexploitedmorewidely.Moreover,thegraphicalmodelformalismprovidesanaturalframeworkforthedesignofnewsystems.Inthispaper,wewill eshoutthisremarkbydiscussingthefollowingtopics:Representation:howcanagraphicalmo

7、delcompactlyrepresentajointprobabilitydistribution?Inference:howcanweecientlyinferthehiddenstatesofasystem,givenpartialandpossiblynoisyobservations?Learning:howdoweestimatetheparametersandstructureofthemodel?Decisiontheory:whathappenswhenitistimetoconvertbeliefsintoactions?

8、Applications:whathasthismachinerybeenusedfor?1P