Probabilistic Graphical Models Inference Message Passing Clique Tree
Probabilistic Graphical Models Inference Message Passing Clique Tree Algorithm & Correctness Daphne Koller
Message Passing in Trees A B C 2 3(C) = 1 2(B) = 1: A, B 2 1(B) = B 2: B, C C 3 2(C) = E D 3 4(D) = 3: C, D D 4: D, E 4 3(D) = Daphne Koller
Correctness 2 3(C) = 1 2(B) = 1: A, B B 2: B, C C 3: C, D D 4: D, E 4 3(D) = Daphne Koller
Clique Tree • Undirected tree such that: – nodes are clusters Ci {X 1, …, Xn} – edge between Ci and Cj associated with sepset Si, j = Ci Cj Daphne Koller
Family Preservation • Given set of factors , we assign each k to a cluster C (k) s. t. Scope[ k] C (k) • For each factor k , there exists a cluster Ci s. t. Scope[ k] Ci Daphne Koller
Running Intersection Property • For each pair of clusters Ci, Cj and variable X Ci Cj there exists a unique path between Ci and Cj for which all clusters and sepsets contain X C 1 C 4 C 7 C 3 C 6 C 5 C 2 Daphne Koller
Running Intersection Property • For each pair of clusters Ci, Cj and variable X Ci Cj, in the unique path between Ci and Cj, all clusters and sepsets contain X C 1 C 4 C 7 C 3 C 6 C 5 C 2 Daphne Koller
More Complex Clique Tree C 1: C, D P(D | C) P(C) D 2: G, I, D G, I P(G | I, D) 3: G, S, I G, S P(I) P(S | I) 4: G, J, S, L P(L | G) P(J | L, S) G, J 5: H, G, J I D G P(H | G, J) S L H J Daphne Koller
Clique Tree Message Passing 3 2(G, I) = 1: C, D 1 2(D)= D 4 3(G, S) = G, I 3: G, S 2: 4: G, I, D G, S, I G, J, S, L 2 3(G, I) = 5 4(G, J) = G, J 5: H, G, J 3 4(G, S)= Daphne Koller
RIP Clique Tree Correctness • If X is eliminated when we pass the message Ci Cj • Then X does not appear in the Cj side of the tree C 1 C 4 C 7 C 3 C 6 C 5 C 2 Daphne Koller
Clique Tree Correctness 4 3(G, S) = 1: C, D P(D | C) P(C) 1 2(D)= D G, I 3: G, S 2: 4: G, I, D G, S, I G, J, S, L P(G | I, D) P(I) P(S | I) P(L | G) P(J | L, S) 5 4(G, J) = G, J 5: H, G, J P(H | G, J) 2 3(G, I) = Daphne Koller
Summary • Belief propagation can be run over a treestructured cluster graph • In this case, computation is a variant of variable elimination • Resulting beliefs are guaranteed to be correct marginals Daphne Koller
END END Daphne Koller
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