In this paper, we address the inverse problem, or the statistical machine learning problem, in Markov random fields with a non-parametric pair-wise energy function with continuous variables. The inverse problem is formulated by maximum likelihood estimation. The exact treatment of maximum likelihood estimation is intractable because of two problems: (1) it includes the evaluation of the partition function and (2) it is formulated in the form of functional optimization. We avoid Problem (1) by using Bethe approximation. Bethe approximation is an approximation technique equivalent to the loopy belief propagation. Problem (2) can be solved by using orthonormal function expansion. Orthonormal function expansion can reduce a functional optimization problem to a function optimization problem. Our method can provide an analytic form of the solution of the inverse problem within the framework of Bethe approximation as a result of variational optimization.
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