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Explore how factorization of a distribution P over a graph implies independencies in P, including Markov networks, independences, and graphical models. Learn about factorization implications, separation in Markov networks, I-map concept, and the relationship between independence and factorization in Hammersley-Clifford theorem.
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Representation Probabilistic Graphical Models Independencies MarkovNetworks
Independence & Factorization • Factorization of a distribution P over a graph implies independencies in P P(X,Y,Z) = P(X) P(Y) X,Y independent • P(X,Y,Z) 1(X,Y) 2(Y,Z) (X Y | Z)
Separation in MNs Definition: X and Y are separated in H given Z if there is no active trail in H between X and Y given Z I(H) = {(X Y | Z) : sepH(X, Y | Z)}
A F D B C E • If P satisfies I(H), we say that H is an I-map (independency map) of P
Factorization Independence • Theorem: If P factorizes over H, then H is an I-map for P A F D B C E
Independence Factorization • Theorem (Hammersley Clifford): For a positive distribution P, if H is an I-map for P, then P factorizes over H
Summary Two equivalent* views of graph structure: • Factorization: H allows P to be represented • I-map: Independencies encoded by H hold in P If P factorizes over a graph H, we can read from the graph independencies that must hold in P (an independency map) * for positive distributions
