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8.1.4-GraphsDistributions

# 8.1.4-GraphsDistributions - Machine Learning Srihari Graphs...

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Machine Learning Srihari 1 Graphs and Distributions Sargur Srihari

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Machine Learning Srihari Topics • I-Maps – I-Map to Factorization – Factorization to I-Map – Perfect Map Knowledge Engineering – Picking Variables – Picking Structure – Picking Probabilities – Sensitivity Analysis 2
Machine Learning Srihari 3 Independencies in a Distribution • Let P be a distribution over X I(P) is set of conditional independence assertions of the form (X Y|Z) that hold in P X Y P(X,Y) x 0 y 0 0.08 x 0 y 1 0.32 x 1 y 0 0.12 x 1 y 1 0.48 X and Y are independent in P , e.g., P(x 1 )=0.48+0.12=0.6 P(y 1 )=0.32+0.48=0.8 P(x 1 ,y 1 )=0.48=0.6 x 0.8 Thus (X Y| ϕ ) I(P)

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Machine Learning Srihari Independencies in a Graph Local Conditional Independence Assertions (starting from leaf nodes): Parents of a variable shield it from probabilistic influence Once value of parents known, no influence of ancestors Information about descendants can change beliefs about a node P ( D , I , G , S , L ) = P ( D ) P ( I ) P ( G | D , I ) P ( S | I ) P ( L | G ) Graph with CPDs is equivalent to a set of independence assertions I ( G ) = {( L I , D , S | G ), ( S D , G , L | I ), ( G S | D , I ), (I D | φ ), ( D I , S | φ )} L is conditionally independent of all other nodes given parent G S is conditionally independent of all other nodes given parent I Even given parents, G is NOT independent of descendant L Nodes with no parents are marginally independent D is independent of non-descendants I and S
Machine Learning Srihari I-MAP • Let G be a graph associated with a set of independencies I(G) • Let P be a probability distribution with a set of independencies I(P) • Then G is an I -map of I if I(G) I(P) From direction of inclusion – distribution can have more independencies than the graph – Graph does not mislead in independencies existing in P 5

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8.1.4-GraphsDistributions - Machine Learning Srihari Graphs...

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