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CS 712Advanced Topics in Aiticial Intelligence: Probabilistic Graphical Models Assignment 1 Spring 2010 Department of Computer Science and...

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CS 712—Advanced Topics in Aitifcial Intelligence: Prob- abilistic Graphical Models Assignment 1 Spring 2010 Department of Computer Science and Engineering Wright State University Due : Monday, May 17 Instructor : Shaojun Wang, Joshi 387, 775-5140, [email protected] Problem 1 There are three parts to this question. Explain both (1) using words and (2) mathematically why is it not valid to have directed cycles in a Bayesian network. Also we described how a Bayesian network is sometimes given a causal interpretation, but how it is best to think of a BN as specifying a factorization rather than ascribing such a casual understanding. Even so, suppose a BN loosely represented causality in some limited circumstances. (3) Would any problems arise in this causal interpretation if a BN was allowed to have directed cycles? Problem 2 In class, we deFned the moralization step as a step that we must perform to convert from a BN into an MR± before we perform (variable) elimination. We stated that moralization was crucial because 1) it keeps the purely graphical elimination procedure from doing something that does not corresponds to summation in directed models (i.e., so that we don’t break apart or factorize p ( c | π c ) in invalid ways in general), and 2) it keeps the MR± from expressing conditional independent properties that the Bayesian network (BN) would not state, but at the cost of loosing some of the independence statements that the BN does state. Regarding conditional independence: 1. Give an example of what a BN would state that the moralization step looses. 2. Give an example of what an MR± would state that would be wrong w.r.t. the BN if moralization was not done. 3. Prove that after Moralization, no CI statements of a BN are violated by the MR± resulting from the moralized BN. I.e., here you need to prove that the MR± obtained by the moralized BN makes no additional conditional independence statement that is not stated by the original BN. 1
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Problem 3 In class we showed that ( F ) = ( G ) = ( L ) = ( P ), but we said it was not the case in general that ( L ) = ( G ), nor was it that ( P ) = ( L ). Let P 0 be the proposition ”( L ) = ( G )”. and P 1 be the proposition ”( P ) = ( L )”. Consider probability distributions p 0 ; p 1 ; p 2 ; p 3 over as many variables as you wish where: For distribution p 0 it must be the case that both P 0 and P 1 are false. For distribution p 1 it must be the case that P 0 is true and P 1 is false. For distribution p 2 it must be the case that P 0 is false and P 1 is true. For distribution p 3 it must be the case that both P 0 and P 1 are true. Your task is to fnd examples of any three out of the above four possibilities, and in each case show why it holds. Problem 4 In class, we stated that if A B | C , then A B | C where A A and B B . 4a) First, suppose that A = B , meaning the sizes of the sets are the same. Suppose also that there is some ordering of A,B , lets say ( A i ,B i ) | A | i =1 ) (where all of A i , and B i are scalars) such that A i B i | C for all i . Does it follow that A B | C ? If so, prove it. If not, ±nd a counterexample. 4b) Next, suppose that it is true that A i B i | C for all A i A and B i B . Does this now imply A B | C ? Again, either prove or ±nd a counterexample. Problem 5 Is this a chordal graph in Figure below? Please explain why. 2
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