I-maps and perfect maps

1. Representation Probabilistic Graphical Models Independencies I-maps andperfect maps

2. 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)

3. 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

4. Minimal I-map • I-map without redundant edges D D I I G G

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

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

7. More imperfect maps • Exactly capture the independencies in P: I(G) = I(P) or I(H) = I(P) X1 X2 Y XOR

8. 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