# C p c p c d e p c 2 p d e f p d e thus we have

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   C p C p C , D , E    p C 2 p D , E , F    p D , E Thus we have reached a fixed point (which we would expect having gone through 2 iterations). Part C Q: Using the maximum likelihood clique potentials learned in part (b). Write out the expression for the joint probability over all the variables. What is the relationship between this expression and the junction tree in part (a). The estimated joint probability distribution is: (30) P IPF A , B , C , D , E , F p A , B , C p C , D , E    p C p D , E , F    p D , E In junction trees the joint probability is equal to the product of the clique potentials divided by the separator potentials. The junction tree obtained from the above graph would be. (31) ABC C CDE DE DEF Thus indeed the joint probability of the junction tree is also written: (32) P JT A , B , C , D , E , F p A , B , C p C , D , E    p C p D , E , F    p D , E Part D [Bonus] Any decomposable model can be represented as a junction tree with maximal cliques as nodes in the junction tree. Hence with potentials over maximal cliques, we get that (33) p X C i p C i    S ij p S ij For MLE, p S p S for any subset of variables S . Hence for MLE, p C i p C i , p S ij p S ij . Therefore with MLE factors, we have (34) p X C i p C i    S ij p S ij 1  Z C i C i C i 8 Author(s) hw5sol.nb12/6/06 Printed by Mathematica for Students
Note that the above equation holds if and only if C i C i are MLE parameters ( if is as above; only if holds because p X , the likelihood function, is maximized with p , and that value of p X should be achieved with MLE parameters of ).

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• Fall '07
• CarlosGustin
• Maximum likelihood, Likelihood function, Marginal distribution, Markov random field, junction tree, Chordal graph

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