1. Bayesian Networks Independencies Representation Probabilistic
Graphical
Models
Independencies
Bayesian 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. d-separation in BNs
Definition: X and Y are d-separated in G given Z if there is no active trail in G between X and Y given Z
no active trail -> independence

4. d-separation examplesG L S
Any node is separated from its
non-descendants givens its parents

5. I-maps
I(G) = {(X  Y | Z) : d-sepG(X, Y | Z)}
If P satisfies I(H), we say that H is an I-map (independency map) of P

6. I-maps P2 P1 G1 G2 I D Prob. i0 d0 0.282 d1 0.02 i1 0.564 0.134 I D
0.42 d1
0.18 i1
0.28
0.12 G1 G2 D I I D
Define I(G)

7. Factorization  Independence: BNs
Theorem: If P factorizes over G, then G is an I-map for P I D G L S
D independent of S

8. Independence  FactorizationG L S
Theorem: If G is an I-map for P, then P factorizes over G

9. Factorization  Independence: BNs
Theorem: If P factorizes over G, then G is an I-map for P I D G L S
D independent of S

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

11. END END END