hw3 - 10708 Graphical Models Homework 3 Due October 27th...

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10708 Graphical Models: Homework 3 Due October 27th, beginning of class October 13, 2006 Instructions : There are six questions on this assignment. Each question has the name of one of the TAs beside it, to whom you should direct any inquiries regarding the question. The last problem involves coding, which should be done in MATLAB. Do not attach your code to the writeup. Instead, copy your implementation to /afs/andrew.cmu.edu/course/10/708/your_andrew_id/HW3 Refer to the web page for policies regarding collaboration, due dates, and extensions. Note : Please put your name and Andrew ID on the first page of your writeup. 1 Triangulation [10 pts] [Khalid] A B C D E F G H Figure 1: Bayes net for question 1 1. Moralize the Bayes net in figure 1. 2. Supply a perfect elimination ordering ( i.e. , one that yields no fill edges). 1

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3. Supply an elimination ordering that yields a triangulated graph with at least 5 nodes in one or more cliques 4. Draw clique trees for the elimination orderings in parts 2 and 3. 2 Clique Tree Factorization [10 pts] [Khalid] 2.1 Prove that the clique beliefs π ( C i ) = P ( C i ) and edge beliefs μ 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 Using the independencies we ask you to show in Question 4, prove that in a clique tree for a BN we can represent the joint distribution by: P ( X ) = i P ( C i ) 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 probabili- ties.) 3 Variable Elimination in Clique Trees [15 pts] [Khalid]
• Fall '07
• CarlosGustin
• Bayesian network, variable elimination, clique tree, Clique Trees

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