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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.

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I-maps and perfect maps

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  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

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