# 22 6 pts now let the distribution of x be given by

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2.2 [6 pts] Now, let the distribution of X be given by some graphical model instance B = ( G , P ). Consider variable X i . What is the minimal subset of the variables, A ⊆ X - { X i } , such that X i is independent of the rest of the variables, X - A ∪ { X i } , given A ? Prove that this subset is necessary and sufficient. 2.3 [5 pts] Show how you could efficiently compute the distribution over a variable X i given some as- signment to all the other variables in the network: P ( X i | x 1 , . . . , x i - 1 , x i +1 , . . . , x n ). Your procedure should not require the construction of the entire joint distribution P ( X 1 , . . . , X n ). Specify the computational complexity of your procedure. 3 [10 pts] Factorization Let G be a bayesian network graph over a set of random variables X and let P be a joint distribution over the same space. Show that if P factorizes according to G , then G is an I - map for P . (Hint: See example on page 86 of Koller and Friedman) 2

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4 [15 pts] Marginalization Burglary Earthquake Alarm TV Nap JohnCall MaryCall Figure 2: Burglar Alarm Network 1. Consider the Burglar Alarm network shown in Figure 2. Construct a Bayesian network over all of the nodes except for Alarm , which is a minimal I - map for the marginal distribution over those variables defined by the above network. Be sure to get all dependencies that remain from the original network. 2. Generalize the procedure you used to solve the above into a node-elimination algorithm. That is, define an algorithm that transforms the structure of G into G 0 such that one of the nodes X ι of G is not in G 0 and G 0 is an I - map of the marginal distribution over the remaining variables as defined by G . 5 [20 pts] Arc Reversal The bayesian network operation of Arc Reversal involves transforming a Bayesian network G containing nodes X and Y as well as arc X Y

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