Daphne Koller

Markov Networks

Independencein Markov Networks

ProbabilisticGraphicalModels

Representation

Daphne Koller

Influence Flow in Undirected Graph

Daphne Koller

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

Daphne Koller

Independences in Undirected Graph

• The independences implied by H

I(H) =

• We say that H is an I-map (independence map) of P if

Daphne Koller

FactorizationP factorizes over H

Daphne Koller

Factorization Independence

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

Daphne Koller

BD

C

A

E

Daphne Koller

Independence Factorization

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

Daphne Koller

Independence Factorization

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

Daphne Koller

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