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1、8GraphicalModelsProbabilitiesplayacentralroleinmodernpatternrecognition.WehaveseeninChapter1thatprobabilitytheorycanbeexpressedintermsoftwosimpleequationscorrespondingtothesumruleandtheproductrule.Alloftheprobabilisticinfer-enceandlearningmanipulationsdisc
2、ussedinthisbook,nomatterhowcomplex,amounttorepeatedapplicationofthesetwoequations.Wecouldthereforeproceedtoformulateandsolvecomplicatedprobabilisticmodelspurelybyalgebraicma-nipulation.However,weshall?ndithighlyadvantageoustoaugmenttheanalysisusingdiagramm
3、aticrepresentationsofprobabilitydistributions,calledprobabilisticgraphicalmodels.Theseofferseveralusefulproperties:1.Theyprovideasimplewaytovisualizethestructureofaprobabilisticmodelandcanbeusedtodesignandmotivatenewmodels.2.Insightsintothepropertiesofthem
4、odel,includingconditionalindependenceproperties,canbeobtainedbyinspectionofthegraph.3593608.GRAPHICALMODELS3.Complexcomputations,requiredtoperforminferenceandlearninginsophis-ticatedmodels,canbeexpressedintermsofgraphicalmanipulations,inwhichunderlyingmath
5、ematicalexpressionsarecarriedalongimplicitly.Agraphcomprisesnodes(alsocalledvertices)connectedbylinks(alsoknownasedgesorarcs).Inaprobabilisticgraphicalmodel,eachnoderepresentsarandomvariable(orgroupofrandomvariables),andthelinksexpressprobabilisticrelation
6、-shipsbetweenthesevariables.Thegraphthencapturesthewayinwhichthejointdistributionoveralloftherandomvariablescanbedecomposedintoaproductoffactorseachdependingonlyonasubsetofthevariables.Weshallbeginbydis-cussingBayesiannetworks,alsoknownasdirectedgraphicalm
7、odels,inwhichthelinksofthegraphshaveaparticulardirectionalityindicatedbyarrows.TheothermajorclassofgraphicalmodelsareMarkovrandom?elds,alsoknownasundirectedgraphicalmodels,inwhichthelinksdonotcarryarrowsandhavenodirectionalsigni?cance.Directedgraphsareusef
8、ulforexpressingcausalrelationshipsbetweenrandomvariables,whereasundirectedgraphsarebettersuitedtoexpressingsoftcon-straintsbetweenrandomvariables.Forthepurposesofsolvinginferenceproblems,itisoftenconvenientto
