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Markov Networks

This article delves into the concepts of independence and factorization in Markov Networks (MNs). It explains how variables X and Y are considered separated given Z if there exists no active trail between them in the graph H. The independence theorem states that if a probability distribution P factorizes over H and separates X, Y given Z, then P satisfies (X ⊥ Y | Z). We explore the implications of H being an independence map (I-map) for P, leading to key insights from Hammersley-Clifford's theorem regarding the structure of positive distributions.

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Markov Networks

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  1. Representation Probabilistic Graphical Models Independencies MarkovNetworks

  2. 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 A F D B C E

  3. Factorization  Independence: MNS Theorem: If P factorizes over H, and sepH(X, Y | Z) then P satisfies (X  Y | Z) A F D B C E

  4. Factorization  Independence: MNs If P satisfies I(H), we say that H is an I-map (independency map) of P Theorem: If P factorizes over H, then H is an I-map of P I(H) = {(X  Y | Z) : sepH(X, Y | Z)}

  5. Independence  Factorization • Theorem (Hammersley Clifford): For a positive distribution P, if H is an I-map for P, then P factorizes over H

  6. 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

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