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

2. Influence Flow in Undirected Graph

3. 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

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
Define I(G)

5. Factorization
P factorizes over H

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

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