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

## TL;DR

This paper presents a novel method for solving the inverse problem in continuous Markov random fields with non-parametric pairwise energy functions, using Bethe approximation and orthonormal function expansion to obtain an analytic solution.

## Contribution

It introduces a new approach combining Bethe approximation and orthonormal function expansion to analytically solve the inverse problem in continuous MRFs, which was previously intractable.

## Key findings

- Provides an analytic solution within Bethe approximation framework.
- Avoids intractable partition function evaluation using Bethe approximation.
- Reduces functional optimization to a manageable function optimization.

## 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.