ps2 - MaryCall SportsOnTV and Naptime which is a minimal...

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Introduction to Probabilistic Graphical Models Problem Set #2 1 Probabilistic Graphical Models, Spring 2009 Problem Set #2 1. (a) Prove that the Weak Union property holds for all distributions P : ( X Y, W | Z ) = ( X Y | Z, W ) (b) Prove the Contraction property for all distributions P : ( X W | Z, Y X Y | Z ) = ( X Y, W | Z ) (c) For positive distributions P , prove the Intersection rule : ( X Y | Z, W X W | Z, Y ) = ( X Y, W | Z ) (d) Provide a counter-example to the Intersection property, in cases when the distribution P is non-positive. 2. (a) Consider the alarm network shown below: Burglary Earthquake Alarm JohnCall MaryCall SportsOnTV Naptime Construct a Bayesian network structure with nodes Burglary, Earthquake, JohnCall,
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Unformatted text preview: MaryCall, SportsOnTV, and Naptime, which is a minimal I-map for the marginal distribution over those variables de±ned by the above network. Be sure to get all dependencies that remain from the original network. (b) Generalize the procedure you used to solve the above into a node-elimination algo-rithm. That is, de±ne an algorithm that transforms the structure of BN into BN such that one of the nodes X i of BN is not in BN and BN is an I-map of the marginal distribution over the remaining variables as de±ned by BN ....
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