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1、MarginalIndependenceandConditionalIndependenceComputerSciencecpsc322,Lecture26(TextbookChpt6.1-2)March,19,2010LectureOverview•RecapwithExample–Marginalization–ConditionalProbability–ChainRule•Bayes'Rule•MarginalIndependence•ConditionalIndependenceourmostbasicandrobustformofknowledgeabou
2、tuncertainenvironments.RecapJointDistribution•3binaryrandomvariables:P(H,S,F)–Hdom(H)={h,h}hasheartdisease,doesnothave…–Sdom(S)={s,s}smokes,doesnotsmoke–Fdom(F)={f,f}highfatdiet,lowfatdietRecapJointDistribution•3binaryrandomvariables:P(H,S,F)–Hdom(H)={h,h}hasheartdisease,doesnothave
3、…–Sdom(S)={s,s}smokes,doesnotsmoke–Fdom(F)={f,f}highfatdiet,lowfatdietffssssh.015.007.005.003h.21.51.07.18RecapMarginalizationffssssh.015.007.005.003h.21.51.07.18P(H,S)P(H,S,Fx)xdom(F)P(H,S)?.02.01P(H)?.28.69P(S)?RecapConditionalProbabilityP(H,S)ssP(H)P(S,H)P(S
4、H)h.02.01
5、.03P(H)h.28.69.97P(S).30.70P(S
6、H)ssP(H
7、S)h.666.333h.29.71RecapConditionalProbability(cont.)P(S,H)P(S
8、H)P(H)Twokeypointswecoveredinpreviouslecture•Wederivedthisequalityfromapossibleworldsemanticsofprobability•Itisnotaprobabilitydistributionsbut…..•Oneforeachconfigurationoftheconditio
9、ningvar(s)RecapChainRuleP(H,S,F)BayesTheoremP(S,H)P(S,H)P(S
10、H)P(H
11、S)P(H)P(S)P(H
12、S)P(S)P(S
13、H)P(H)LectureOverview•RecapwithExampleandBayesTheorem•MarginalIndependence•ConditionalIndependenceDoyoualwaysneedtoreviseyourbeliefs?……whenyourknowledgeofY’svaluedoesn’taffectyourbeliefintheval
14、ueofXDEF.RandomvariableXismarginalindependentofrandomvariableYif,forallxdom(X),ydom(Y),ikP(X=x
15、Y=y)=P(X=x)ikiMarginalIndependence:Example•XandYareindependentiff:P(X
16、Y)=P(X)orP(Y
17、X)=P(Y)orP(X,Y)=P(X)P(Y)•ThatisnewevidenceY(orX)doesnotaffectcurrentbeliefinX(orY)•Ex:P(Toothache,Catch,Cav
18、ity,Weather)=P(Toothache,Catch,Cavity.•JPDrequiringentriesisreducedtotwosmallerones(and)InourexampleareSmokingandHeartDiseasemarginallyIndependent?Whatourprobabilitiesaretellingus….?P(H,S)ssP(H)h.02.01.03h.28.69.97P(S).30.70P(S
19、H)ssh.666.334h.29.71LectureOverview•RecapwithE