final-07

final-07 - CS228 Final 1 CS 228, Winter 2007 Final Handout...

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Unformatted text preview: CS228 Final 1 CS 228, Winter 2007 Final Handout #17 You have 24 hours to complete this exam. The exam is given out at noon, and due at noon (12:00 pm) one day after you pick it up. The exam will be handed out and collected in Gates 120 (the Fishbowl). This exam is long and difficult, and we do not expect everyone to finish all of the questions. Be sure to use good test taking skills and attack the easier problems first, spend more time on questions worth more points, and generally pay attention to how you spend your time. Also, you are welcome (and we expect you) to use or refer algorithms from the reader when appropriate, without having to rederive or explain them. Furthermore, please use standard notation from the reader (when possible) and clearly define any terms you introduce. Algorithm answers should be provided in the form of pseudocode (with explanations where necessary), and your answers will be easier to grade (read: you will get higher grades) if you use proper spacing and layout of your answers on the page. We have provided approximate times with each of the problems to give you a rough estimate of how long we think it might take (in time and pages not including diagrams). 1. [10 points] Representation (a) [5 points] Let G be a Bayesian network with no immoralities. Let H be a Markov network with the same skeleton as G . Show that H is an I-map of I ( G ). That is, prove Corollary 5.7.4 Corollary 5.5.4 in the version of Chapter 5 posted on Coursework for the case in which G has no immoralities. (You may not simply assume this corollary or use theorems that follow it in the readings.) (b) [5 points] Let X and Y be variables in a Bayesian network G , and suppose that neither is a parent of the other. Let W = Pa X Pa Y . Prove that ( X Y | W ) I ( G ). Estimate: 1 page, 1 hour 2. [20 points] Sampling on a Tree Suppose we have a distribution P ( X , E ) over two sets of variables X and E . Our distri- bution is represented by a nasty Bayes Net with very dense connectivity, and our sets of variables X and E are spread arbitrarily throughout the network. In this problem our goal is to use the sampling methods we learned in class to estimate the posterior probability P ( X = x | E = e ). More specifically, we will use a tree-structured Bayes Net as the proposal distribution for use in the importance sampling algorithm. (a) [1 points] For a particular value of x and e , can we compute P ( x | e ) exactly, in a tractable way? Can we sample directly from the distribution P ( X | e )? Can we compute P ( x | e ) = P ( x , e ) exactly, in a tractable way? For each question, provide a Yes/No answer and a single sentence explanation or description....
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final-07 - CS228 Final 1 CS 228, Winter 2007 Final Handout...

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