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Unformatted text preview: CS228 Problem Set #2 Solutions 1 CS 228, Winter 2007 Problem Set #2 Solutions Handout #12 1. Collapsed Gibbs Sampling In this problem we will explore a few variations of Col lapsed Gibbs sampling, as described in Section 10.5.2 (which you should read carefully before starting this problem). Consider a Bayesian Network such as the one in Figure 1 representing how a small towns voters V = { V 1 , . . ., V k } are affected by the quality of two sets of volunteers: K = { K 1 , . . . , K n } for the Kerry campaign and B = { B 1 , . . ., B m } for the Bush campaign. Each party talks to each voter exactly once via a particular one of its volunteers. Our goal is to estimate the distribution over the quality of the volunteers given how the population voted, P ( B , K  v ). You may assume the following: The range of volunteer quality is discretized such that for any volunteer X B K we have that  V al ( X )  = d . The voter makes his or her decision known freely to anyone, giving us the evidence V = v . K K B K n V 1 2 V V k V k1 1 B 2 B 2 1 m Figure 1: Network for collapsed gibbs sampling (a) [12 points] The first sampling algorithm we explore computes a set of collapsed particles, each of which specifies an assignment to the B variables, and a closedform distribution over the K variables: ( b [ ] , P ( K ) ) , where P ( K ) = P ( K  b [ ] , v ). i. [4 points] Show how to efficiently compute, in closed form, the distribution P ( K )....
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 Winter '09

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