I-maps and perfect maps
description
Transcript of I-maps and perfect maps
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