10-708-05F - 10708 Graphical Models Homework 1 Due October...

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Unformatted text preview: 10708 Graphical Models: Homework 1 Due October 3rd, beginning of class September 20, 2005 1 [15 pts] Conditional Probability 1.1 [5 pts] Let X , Y , Z be three disjoint sets of variables such that S = X ∪ Y ∪ Z . Prove that P | = ( X ⊥ Y|Z ) if and only if we can write P in the form: P ( S ) = f ( X , Z ) g ( Y , Z ) 1.2 [3 pt] Is it possible for both f and g above to be probability distributions over their respective sets of variables? Formally, is it possible for every distribution P over ( X ∪ Y ∪ Z ) with the independency above, to be expressed as a product of a distribution over ( X ∪ Z ) and a distribution over ( Y ∪ Z )? Justify your answer. (Hint: Assume the variables are binary; look at the marginal probability of Z ) 1.3 [4 pts] Prove or disprove (by providing a counter-example) each of the following properties of inde- pendence: 1. ( X ⊥ Y, W | Z ) implies ( X ⊥ Y | Z ). 2. ( X ⊥ Y | Z ) and ( X, Y ⊥ W | Z ) imply ( X ⊥ W | Z ). 3. ( X ⊥ Y, W | Z ) and ( Y ⊥ W | Z ) imply ( X, W ⊥ Y | Z ). 4. ( X ⊥ Y | Z ) and ( X ⊥ Y | W ) imply ( X ⊥ Y | Z, W ). 1.4 [3 pt] Provide an example of a distribution P ( X 1 , X 2 , X 3 ) where for each i 6 = j , we have that ( X i ⊥ X j ) ∈ I ( P ), but we also have that ( X 1 , X 2 ⊥ X 3 ) / ∈ I ( P ). 1 2 [15 pts] Graph Independencies 2.1 [4 pts] X1 X2 X3 X6 X8 X4 X5 X7 Figure 1: Graphical Model for Prob. 2 Let X = { X 1 , . . . , X n } be a random vector with distribution given by the graphical model in Figure 1. Consider variable X 1 . What is the minimal subset of the variables, A ⊆ X - {...
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10-708-05F - 10708 Graphical Models Homework 1 Due October...

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