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Understanding Independence in Probabilistic Graphical Models

Explore the concepts of independence in Markov networks and graphical models, including separation, influence flow, and I-maps. Learn the theorems linking factorization to independence and vice versa, along with the practical implications for modeling and inference.

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Understanding Independence in Probabilistic Graphical Models

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  1. Representation Probabilistic Graphical Models Markov Networks Independencein Markov Networks

  2. Influence Flow in Undirected Graph

  3. Separation in Undirected Graph • A trail X1—X2—… —Xk-1—Xk is activegiven Z • X and Y are separated in H given Z if

  4. Independences in Undirected Graph • The independences implied by H I(H) = • We say that H is an I-map (independence map) of P if

  5. Factorization P factorizes over H

  6. Factorization  Independence Theorem: If P factorizes over H then H is an I-map for P

  7. A D B C E

  8. Independence  Factorization Theorem: If H is an I-map for P then P factorizes over H

  9. Independence  Factorization Hammersley-Clifford Theorem: If H is an I-map for P, and P is positive, then P factorizes over H

  10. Summary • Separation in Markov network H allows us to “read off” independence properties that hold in any Gibbs distribution that factorizes over H • Although the same graph can correspond to different factorizations, they have the same independence properties

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