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In this paper, we address the inverse problem, or the statistical machinelearning problem, in Markov random fields with a non-parametric pair-wiseenergy function with continuous variables. The inverse problem is formulated bymaximum likelihood estimation. The exact treatment of maximum likelihoodestimation is intractable because of two problems: (1) it includes theevaluation of the partition function and (2) it is formulated in the form offunctional optimization. We avoid Problem (1) by using Bethe approximation.Bethe approximation is an approximation technique equivalent to the loopybelief propagation. Problem (2) can be solved by using orthonormal functionexpansion. Orthonormal function expansion can reduce a functional optimizationproblem to a function optimization problem. Our method can provide an analyticform of the solution of the inverse problem within the framework of Betheapproximation.

Solving Non-parametric Inverse Problem in Continuous Markov Random Field using Loopy Belief Propagation