Be a random vector with distribution given by the

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be a random vector with distribution given by the graphical model in Figure 2. Consider variable X 1 . What is the minimal subset of the variables, A ⊆ X -{ X 1 } , such that X 1 is independent of the rest of the variables, X - A ∪ { X 1 } , givan A ? Justify your answer. 2
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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
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be a random vector with distribution given by the graphical...

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