ps2-practice-sol (2)

ps2-practice-sol (2) - CS228 Practice Questions 1 CS 228...

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CS228 Practice Questions 1 CS 228, Winter 2010 Practice Questions 1. Collapsed Gibbs Sampling In this problem we will explore a few variations of Collapsed Gibbs sampling. Consider a Bayesian Network such as the one in Figure 1 representing how a small town’s 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 k-1 1 B 2 B 2 1 m Figure 1: Network for distributional gibbs sampling (a) [12 points] The first sampling algorithm we explore computes a set of distributional particles, each of which specifies an assignment to the B variables, and a closed-form 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 ). Answer: We first observe that ( K i K j | B , V ) since there is no active path between any K i and K j pair given B and V . Thus we have, P ( K ) = P ( K | b [ ] , v ) = n productdisplay i =1 P ( K i | b [ ] , v ) = n productdisplay i =1 P ( K i | b K i [ ] , v K i ) where we have written b K i [
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