1. I-maps and perfect maps
Representation
Probabilistic
Graphical
Models
Independencies
I-maps and perfect 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 I D G I D G

5. 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

6. 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

7. 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

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

9. END END END