8.1.5-Reasoning&D-Separation

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1、MachineLearning!!!!!SrihariReasoningPatternsandD-SeparationSargurSriharisrihari@cedar.buffalo.edu1MachineLearning!!!!!SrihariTopics•ReasoningPatterns–CausalandEvidentialReasoning•D-separation–DirectConnection–IndirectConnection•SoundnessandCompleteness•AlgorithmforD-separation•I-Equiv

2、alence•MarkovBlanket2MachineLearning!!!!!SrihariBayesianNetwork:StudentModelGraphandCPDsVal(I)={i0=lowintelligence,d0d1i0i1i1=highintelligence}0.60.40.70.3DifficultyIntelligenceVal(D)={d0=easy,d1=hard}g1g2g3GradeSATi0,d00.30.40.3123i0,d10.050.250.7Val(G)={g=A,g=B,g=C}i0,d00.90.080.02s0

3、s1Letteri0,d10.50.30.2i00.950.05i10.20.8Val(S)={s0=low,s1=high)01llg10.10.9g20.40.601g20.990.01Val(L)={l=weak,l=strong}P(D,I,G,S,L)=P(D)P(I)P(G

4、D,I)P(S

5、I)P(L

6、G)Chainrulefor10210102101102P(i,d,g,s,l)=P(i)P(d)P(g

7、i,d)P(s

8、i)P(l

9、g)Bayesiannetwork3=0.3⋅0.6⋅0.08⋅0.8⋅0.4=0.004608MachineLearn

10、ing!!!!!SrihariLocalIndependenciesinaBN•ABNGisadirectedacyclicgraphwhosenodesrepresentrandomvariablesX,..,X.in•LetPa(X)denoteparentsofXinGii•LetNon-descendants(X)denotevariablesinGthatiarenotdescendantsofXi•ThenGencodesthefollowingsetofconditionalindependenceassumptionsdenotedI(G)l•Fo

11、reachX:(X⊥NonDescendantsX

12、Pa(X))iiii•AlsoknownasLocalMarkovIndependencies4MachineLearning!!!!!SrihariExampleofLocalIndependenciesd0d1i0i10.60.40.70.3DifficultyIntelligence•Graphisequivalenttoag1g2g3setoflocalindependenceGradeSATi0,d00.30.40.3i0,d10.050.250.7assertionsi0,d00.90.080.02s0

13、s1Letteri0,d10.50.30.2i00.950.05i10.20.8l0l1g10.10.9g20.40.6g20.990.01•LocalConditionalIndependenceAssertions(startingfromleafnodes):I(G)={(L⊥I,D,S

14、G),LisconditionallyindependentofallothernodesgivenparentG(S⊥D,G,L

15、I),SisconditionallyindependentofallothernodesgivenparentI(G⊥S

16、D,I),Eve

17、ngivenparents,GisNOTindependentofdescendantL(I⊥D

18、φ),Nodeswithnoparentsaremarginallyindependent(D⊥I,S

19、φ)}Disindependentofnon-descendantsIandS•Parentsofavariableshielditfromprobabilisticinfluence•Oncevalueofparentsknown,noinfluenceofancestors•Informationaboutdescendantscanchangebeliefsa

20、bouta