Title: Solving Non-parametric Inverse Problem in Continuous Markov Random Field using Loopy Belief Propagation,
Abstract: 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.