Clique Tree & Independence Inference Probabilistic Graphical Models Message Passing Clique Tree & Independence

RIP and Independence For an edge (i,j) in T, let: W<(i,j) = all variables that appear only on Ci side of T W<(j,i) = all variables that appear only on Cj side Variables on both sides are in the sepset Si,j Theorem: T satisfies RIP if and only if, for every (i,j)

RIP and Independence 1: C,D 2: G,I,D 3: G,S,I 5: H,G,J 4: G,J,S,L D

RIP and Independence Theorem: T satisfies RIP if and only if, for every edge (i,j) C D I S G L J H 1: C,D 2: G,I,D 3: G,S,I 5: H,G,J 4: G,J,S,L D G,I G,S G,J

Implications Each sepset needs to separate graph into two conditionally independent parts A1,1 A1,2 A1,3 A1,4 A2,1 A2,2 A2,3 A2,4 A3,1 A3,2 A3,3 A3,4 A4,1 A4,2 A4,3 A4,4 A1 A2 Ak … B1 B2 Bm

Summary Correctness of clique tree inference relies on running intersection property Running intersection property implies separation in original distribution Implies minimal complexity incurred by any clique tree: Related to minimal induced width of graph

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