# hw4 - 10708 Graphical Models Homework 4 Due November 15th...

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10708 Graphical Models: Homework 4 Due November 15th, beginning of class October 27, 2006 Instructions : There are six questions on this assignment. Each question has the name of one of the TAs beside it, to whom you should direct any inquiries regarding the question. The last problem involves coding. Do not attach your code to the writeup. Instead, copy your implementation to /afs/andrew.cmu.edu/course/10/708/your_andrew_id/HW4 Refer to the web page for policies regarding collaboration, due dates, and extensions. Note : Please put your name and Andrew ID on the first page of your writeup. 1 Markov Network Representations [5 pts] [Khalid] Figure 1 is a Markov Random Field where the potentials are defined on all cliques of three variables. A B C D 1 1 8 1 1 22 1 0 14 0 1 1 12 0 1 0 0 1 1 7 0 0 1 1 0 1 0 5 0 15 0 0 Ψ (A,B,C) C B A 1 1 1 1 1 2 1 0 15 0 1 1 13 0 1 0 0 1 1 11 0 0 9 1 0 1 0 3 0 6 0 0 Ψ (B,C,D) D C B Figure 1: A chordal (triangulated) Markov network (a) Convert the triangle graph on ( A, B, C ) with potential Ψ( A, B, C ) into a pairwise Markov Random Field by introducing a new variable X. Show the graph, as well as the node and edge potentials in table form ( i.e. , compute the values of the potentials in the pairwise MRF) 1

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(b) Convert the graph on ( A, B, C, D ) with potentials Ψ( A, B, C ) and Ψ( B, C, D ) into a pairwise Markov Random Field. Is the graph chordal ? ( Note : You do not have to compute the pairwise MRF potentials in your solution). 2 Hammersley-Clifford [10 pts] [Ajit] Complete the analysis of Example 5.4.3 (Koller & Friedman, pg 199), showing that the distribution P defined in the example does not factorize over H . ( Hint : Use a proof by contradiction). 3 Importance Sampling [20 pts] [Khalid] To do this question you need to read (Koller & Friedman, 10.2.2). The likelihood weighting of an importance sampler is defined as w ( x ) = P ( x ) /Q ( x ) where P is the distribution we want to sample from and Q is a proposal distribution. (a) Why is computing the probability of a complete instantiation of the variables in a Markov Random Field computationally intractable ? Your answer should be brief.
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