1. Markov Networks Independencies Representation Probabilistic Graphical
Models
Independencies
Markov Networks

2. Independence & Factorization
P(X,Y,Z) = P(X) P(Y)
X,Y independent
P(X,Y,Z)  1(X,Y) 2(Y,Z)
(X  Y | Z)
Factorization of a distribution P over a graph implies independencies in P
Provides rigorous semantics for flow of influence arguments

3. Separation in MNs
Definition: X and Y are separated in H given Z if there is no active trail in H between X and Y given Z
I(H) = {(X  Y | Z) : sepH(X, Y | Z)}
no active trail -> independence

4. If P satisfies I(H), we say that H is an I-map (independency map) of PB C E
If P satisfies I(H), we say that H is an I-map (independency map) of P

5. Factorization  Independence
Theorem: If P factorizes over H, then H is an I-map for P B D C A E F

6. Independence  Factorization
Theorem (Hammersley Clifford):
For a positive distribution P, if H is an I-map for P, then P factorizes over H

7. Summary Two equivalent* views of graph structure:
Factorization: H allows P to be represented
I-map: Independencies encoded by H hold in P
If P factorizes over a graph H, we can read from the graph
independencies that must hold in P (an independency map)
* for positive distributions

8. END END END