ps2-08

# ps2-08 - CS228 Problem Set #2 1 CS 228, Winter 2008 Problem...

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CS228 Problem Set #2 1 CS 228, Winter 2008 Problem Set #2 We have provided approximate lengths with each of the problems to give you a rough estimate of how long we think each answer might be not including diagrams. These are meant to be guidelines only to make sure that answers are concise and readable (for diagrams, the larger the better). 1. D-Separation [22 points] In this problem, you will show that the set of variables in a clique tree separator d-separates the graph into two conditionally independent pieces. (a) [7 points] For an undirected graph H , recall that we say sep H ( X ; Y | Z ) (i.e., X and Y are separated given evidence Z in H ) whenever all paths between variables in X and variables in Y are blocked by some variables in Z . Prove that if we have a moralized graph H derived from an original Bayesian network G then sep H ( X ; Y | Z ) implies d - sep G ( X ; Y | Z ). This is a generalization of your result from Problem 5 on the previous problem set, and will be used in the next part. (b) [15 points] Now, assume that we perform variable elimination on our Bayesian net- work G , using the Sum-Product-Variable-Elimination algorithm (see section 10.2.1.2). This deﬁnes an induced graph I where each intermediate factor ψ during the elimi- nation process deﬁnes a clique (see section 11.1). Furthermore, as described in class this deﬁnes a clique tree T where each intermediate factor ψ is represented as a clique in the tree and therefore as a subset of a maximal clique in I . Let S be a clique separator in the clique tree T . Let X be the set of all variables mentioned in T on one side of the separator and let Y be the set of all variables mentioned on the other side of the separator. Prove that sep I ( X ; Y | S ) and con- clude (using the above result) that d - sep G ( X ; Y | S ). (Hint: Remember the running intersection property.) Estimate: 2 pages 2. Message Passing [14 points] Let B be a Bayesian Network over a set of n random variables X and F be the set of factors corresponding to the CPDs of B . Suppose that we then construct a clique tree T by choosing a variable elimination ordering and ﬁnding the maximal cliques in the induced graph I F , . Our task is to optimize the message passing that takes place as we calibrate T

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ps2-08 - CS228 Problem Set #2 1 CS 228, Winter 2008 Problem...

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