Daphne Koller

Bayesian Networks

Factorization & Independence

ProbabilisticGraphicalModels

Representation

Daphne Koller

Dual View

Daphne Koller

Independence Assumptions in G

• The independencies implied by G I(G) =

Daphne Koller

G and PWe say that G is an I-map (independence map) of P if

Daphne Koller

I-maps

I D Prob

i0 d0 0.42

i0 d1 0.18

i1 d0 0.28

i1 d1 0.12

I D Prob.i0 d0 0.282i0 d1 0.02 i1 d0 0.564i1 d1 0.134

P1P2

Daphne Koller

Factorization Independence

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

ID

G

L

S

Daphne Koller

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

Daphne Koller

Independence Factorization

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

G

L

S

Daphne Koller

ID

G

L

S

Daphne Koller

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

Daphne Koller

END END END