MLE-induced Likelihood for Markov Random Fields

TL;DR

This paper introduces a novel likelihood approximation method for Markov random fields that combines marginal likelihood functions via copulas, outperforming traditional methods in accuracy and computational efficiency.

Contribution

The paper proposes a new likelihood approximation technique for MRFs using marginal likelihoods and copulas, addressing intractability issues more effectively than existing methods.

Findings

Our approach shows superior numerical performance.

It maintains efficiency as MRF size increases.

It outperforms Laplace and pseudolikelihood methods.

Abstract

Due to the intractable partition function, the exact likelihood function for a Markov random field (MRF), in many situations, can only be approximated. Major approximation approaches include pseudolikelihood and Laplace approximation. In this paper, we propose a novel way of approximating the likelihood function through first approximating the marginal likelihood functions of individual parameters and then reconstructing the joint likelihood function from these marginal likelihood functions. For approximating the marginal likelihood functions, we derive a particular likelihood function from a modified scenario of coin tossing which is useful for capturing how one parameter interacts with the remaining parameters in the likelihood function. For reconstructing the joint likelihood function, we use an appropriate copula to link up these marginal likelihood functions. Numerical investigation suggests the superior performance of our approach. Especially as the size of the MRF increases, both the numerical performance and the computational cost of our approach remain consistently satisfactory, whereas Laplace approximation deteriorates and pseudolikelihood becomes computationally unbearable.