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Unformatted text preview: 10708 Graphical Models: Homework 2 Due October 17th, beginning of class October 3, 2005 1 [15 pts] Clique Tree I-maps In order to formalize the relationship between clique trees and Bayesian Networks, in this question you will prove that if P factorizes according to a Bayesian Network, then any clique tree T for this BN is an I-map for P . Specifically; In a clique tree, consider a separator S ij between two cliques C i and C j . Let X be any set of variables in the C i side of the tree, and Y be any set of variables in the C j side of the tree. Prove that P | = ( X Y | S ij ). 2 [15 pts] Clique Tree Factorization 2.1 [7 pts] Prove that the clique beliefs ( C i ) = P ( C i ) and edge beliefs ij ( S ij ) = P ( S ij ) form a fixed point for the belief propagation algorithm for a clique tree, i . e . , if we start BP with these beliefs, no messages will change them. 2.2 [8 pts] Using the independencies you showed in Question 1, prove that in a clique tree for a BN we can represent the joint distribution by: P ( X ) = producttext i P ( C i ) producttext ij P ( S ij ) . You should not prove by corollary from the correctness of BP in clique trees. (Hint: combine the chain rule of probabilities with the definition of conditional probabilities.) 1 3 [6 pts] Triangulation 3.1 [3 pts] A C F G E B D Figure 1: Moralized Graph for Prob. 2.1 Consider the graph in Figure 1.Consider the graph in Figure 1....
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- Fall '07