Independence in Markov Networks Representation Probabilistic Graphical Models Markov Networks Independence in Markov Networks

Influence Flow in Undirected Graph

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

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

Factorization P factorizes over H

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

B D C A E

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

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

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