Independence in Markov Networks

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