Factorization & Independence Representation Probabilistic Graphical Models Bayesian Networks Factorization & Independence

Dual View

Independence Assumptions in G The independencies implied by G I(G) = Define I(G)

G and P We say that G is an I-map (independence map) of P if Define I(G)

I-maps P2 P1 I D Prob. i0 d0 0.282 d1 0.02 i1 0.564 0.134 I D Prob i0 0.42 d1 0.18 i1 0.28 0.12 Define I(G)

Factorization  Independence Theorem: If P factorizes over G then G is an I-map for P I D G L S

P(D,I,G,S,L) = P(D) P(I) P(G | I,D) P(L | G) P(S | I)

Independence  Factorization Theorem: If G is an I-map for P then P factorizes over G I D G L S

I D G L S

Summary d-separation allows us to use G to read off independencies that must hold in any distribution P that factorizes over G If the d-separation independencies hold in P, it must be representable as a BN over G

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