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ps3 - Z Provide a transformation G of G such that we can...

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Introduction to Probabilistic Graphical Models Problem Set #3 1 Probabilistic Graphical Models, Spring 2009 Problem Set #3 1. (a) Prove the following theorem: Let G 1 and G 2 be two graphs over X . If they have the same skeleton and the same set of v-structures then they are I-equivalent. (b) How many networks are equivalent to the simple directed chain X 1 X 2 . . . X n ? 2. In this question, we will consider the sensitivity of a particular query P ( X | Y ) to the CPD of a particular node Z . Let X and Z be nodes, and Y be a set of nodes. Provide a sound and “complete” criterion for determining when the result of the query P ( X | Y ) is not aFected by the choice of the CPD P ( Z | Pa Z ). More precisely, consider two networks B 1 and B 2 that have identical graph structure G and identical CPDs everywhere except at the node
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Unformatted text preview: Z . Provide a transformation G of G such that we can test whether P B 1 ( X | Y ) = P B 2 ( X | Y ) using a single d-separation query on G . Note that Z may be the same as X , and that Z may also belong to Y . Your criterion should always be sound, but only be complete in the same sense that d-separation is complete, i.e., for “generic” CPDs. Hint: Although it is possible to do this problem using laborious case analysis, it is signi±-cantly easier to think of how a d-separation query on a transformed graph G can be used to detect whether a perturbation on the CPD P ( Z | Pa Z ) makes a diFerence to P ( X | Y )....
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