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# Tree-reweighted Belief Propagation Algorithms and Approximate ML Estimation by Pseudo-Moment Matching

*Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics*, PMLR R4:308-315, 2003.

#### Abstract

In previous work [10] we presented a class of upper bounds on the log partition function of an arbitrary undirected graphical model based on solving a convex variational problem. Here we develop a class of local message-passing algorithms, which we call tree-reweighted belief propagation, for efficiently computing the value of these upper bounds, as well as the associated pseudomarginals. We also consider the uses of our bounds for the problem of maximum likelihood (ML) parameter estimation. For a completely observed model, our analysis gives rise to a concave lower bound on the log likelihood of the data. Maximizing this lower bound yields an approximate ML estimate which, in analogy to the moment-matching of exact ML estimation, can be interpreted in terms of pseudo-moment-matching. We present preliminary results illustrating the behavior of this approximate ML estimator.