hw3sol - 10708 Graphical Models: Homework 3 Solutions 1...

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10708 Graphical Models: Homework 3 Solutions 1 Triangulation 1. The moralized Bayes net (figure 1) is produced by marrying the parents and dropping direction on DAG edges. A B C D E F G H Figure 1: Moralized Bayes Net 2. A,B,G,H,D,E,F,C 3. C,A,B,G,H,D,E,F 4. The clique trees are in figures 2 and 3. 2 Clique Tree Factorization 2.1 Let us prove that if the clique tree is in the state π ( C i ) = P ( C i ) , μ ij ( S ij ) = P ( S ij ) , (1) 1
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ABC EFH 0 CEF 1 DEC 1 2 DEG 0 1 1 2 1 2 Figure 2: Clique Tree for part 2 ABCDEF DEG 2 EFH 2 1 Figure 3: Clique Tree for part 3 then a new message for any edge and any direction i j will be equal to μ ij and will not change the potential of C j . By definition of belief propagation, δ i j = X C i \ S ij π i = X C i \ S ij P ( C i ) = P ( S ij ) = μ ij (2) so the edge belief for edge i - j does not change, and because π new j = π j δ i j μ ij = π j × 1 = π j (3) the belief of clique j does not change either. 2.2 Consider an edge i - j , where clique C i is a leaf in the clique tree. Let W i = C i \ S ij . By the chain rule, P ( X ) = P ( W i | X \ W i ) P ( X \ W i ) (4) and by the independence property from question 4, P ( W i | X \ W i ) = P ( W i | S ij ) = P ( W i , S ij ) P ( S ij ) = P ( C i ) P ( S ij ) . (5) 2
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Thus, P ( X ) = P ( C i ) P ( S ij ) P ( X \ W i ) . (6) Now apply the same reasoning to any leaf of C \ C i , where C is the original clique tree, until there are no more cliques (each clique will become a leaf at some point in time, when all but one of its neighbors are eliminated). Each clique C j will contribute a factor P ( C j ) P ( S jk ) (7) where j - k is one of the edges involving clique C j , to result in P ( X ) = Q i P ( C i ) Q ( i,j ) E P ( S ij ) (8) (each P ( S ij ) will be contributed only once, when the first of two corresponding cliques is eliminated). 3
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3) Variable Elimination in Clique Trees Q: Consider a chain graphical model with the structure X 1 - X 2 - - X n , where X i takes on one of d possible assignments. You can form the following clique tree for this GM: C 1 - C 2 - - C n - 1 , where Scope @ C i D = 8 X i , X i + 1 < . You can assume that this clique tree has already been calibrated. Using this clique tree, we can directly obtain
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hw3sol - 10708 Graphical Models: Homework 3 Solutions 1...

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