Daphne Koller

Independencies

BayesianNetworks

ProbabilisticGraphicalModels

Representation

Daphne Koller

Independence & Factorization

• Factorization of a distribution P over a graph implies independencies in P

P(X,Y,Z) 1(X,Y) 2(Y,Z)

P(X,Y,Z) = P(X) P(Y) X,Y independent

(X Y | Z)

Daphne Koller

d-separation in BNsDefinition: X and Y are d-separated in G given

Z if there is no active trail in G between X and Y given Z

Daphne Koller

d-separation examplesID

G

L

S

Any node is separated from its non-descendants givens its parents

Daphne Koller

I-maps

• If P satisfies I(H), we say that H is an I-map (independency map) of P

I(G) = {(X Y | Z) : d-sepG(X, Y | Z)}

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

G1

P2

IDID

P1

G2

Daphne Koller

Factorization Independence: BNs

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

ID

G

L

S

Daphne Koller

Independence Factorization

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

ID

G

L

S

Daphne Koller

Factorization Independence: BNs

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

ID

G

L

S

Daphne Koller

SummaryTwo 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)

Daphne Koller

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