I-maps and perfect maps Representation Probabilistic Graphical Models Independencies I-maps and perfect maps

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)

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

Minimal I-map I-map without redundant edges I D G I D G

MN as a perfect map Exactly capture the independencies in P: I(G) = I(P) or I(H) = I(P) I D G I D G

BN as a perfect map Exactly capture the independencies in P: I(G) = I(P) or I(H) = I(P) B D C A B D C A B D C A B D C A

More imperfect maps Exactly capture the independencies in P: I(G) = I(P) or I(H) = I(P) X1 X2 Y Prob 0.25 1 X1 X2 Y XOR

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