2 figure 2 graphical model for prob 2 22 8 pts now

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Figure 2: Graphical Model for Prob. 2 2.2 [8 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. (Hint: Think about the variables that X i cannot possibly be conditionally independent of, and then think some more) 2.3 Extra Credit [8 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 Marginalization [15 pts] 1. Consider the Burglar Alarm network shown in Figure 3. 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.
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