1. Factorization & Independence
Representation
Probabilistic
Graphical
Models
Bayesian Networks
Factorization & Independence

2. Dual View

3. Independence Assumptions in G
The independencies implied by G
I(G) =
Define I(G)

4. G and P We say that G is an I-map (independence map) of P if
Define I(G)

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

6. Factorization  Independence
Theorem: If P factorizes over G then G is an I-map for P I D G L S

7. P(D,I,G,S,L) = P(D) P(I) P(G | I,D) P(L | G) P(S | I)

8. Independence  Factorization
Theorem: If G is an I-map for P then P factorizes over G I D G L S

9. I D G L S

10. Summary
d-separation allows us to use G to read off independencies that must hold in any distribution P that factorizes over G
If the d-separation independencies hold in P, it must be representable as a BN over G

11. END END END