Daphne Koller

Independencies

I-maps andperfect maps

ProbabilisticGraphicalModels

Representation

Daphne Koller

Capturing Independencies in P

• P factorizes over G G is an I-map for P:

• But not always vice versa: there can be independencies in I(P) that are not in I(G)

Daphne Koller

Want a Sparse Graph• If the graph encodes more

independencies– it is sparser (has fewer parameters)– and more informative

• Want a graph that captures as much of the structure in P as possible

Daphne Koller

Minimal I-map• I-map without redundant edges

ID

G

ID

G

Daphne Koller

MN as a perfect map• Exactly capture the independencies in P:

I(G) = I(P) or I(H) = I(P)

ID

G

ID

G

Daphne Koller

BN as a perfect map• Exactly capture the independencies in P:

I(G) = I(P) or I(H) = I(P)

BD

C

A

BD

C

A

BD

C

A

BD

C

A

Daphne Koller

More imperfect maps• Exactly capture the independencies in P:

I(G) = I(P) or I(H) = I(P)

X2X1

Y XOR

X1 X2Y Prob

0 0 0 0.25

0 1 1 0.25

1 0 1 0.25

1 1 0 0.25

Daphne Koller

Summary• Graphs that capture more of I(P) are

more compact and provide more insight

• But it may be impossible to capture I(P) perfectly (as a BN, as an MN, or at all)

• BN to MN: lose the v-structures• MN to BN: add edges to undirected

loops