HW2-2 - Probabilistic Graphical Models Homework 2 Due April...

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Probabilistic Graphical Models: Homework 2 Due April 28th, (Ordibehesht 9th), beginning of class April 17, 2013 Instructions : There are 3 questions on this assignment. Please submit your homework in the same order they appeared in the homework. 1 Conditional Probability 1.1 [4 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 [5 pts] 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 ( Y∪Z )? Justify your answer. ( Hint : look at the marginal probability of Z ; you may assume that the variables are binary if you wish.) 1.3 [3 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 ).
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  • Spring '13
  • Dr.ZAre
  • Probability theory, pts, 3 pts, 4 pts, graphical model

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