Minimum Conditional Description Length Estimation for Markov Random Fields

TL;DR

This paper introduces the Minimum Conditional Description Length (MCDL) method for estimating parameters in subsets of Markov random fields, leveraging known graph edges and boundary conditions to improve local parameter estimation.

Contribution

The paper proposes a novel MCDL approach for local parameter estimation in Markov random fields, connecting it with the Maximum Pseudo-Likelihood method for single-configuration scenarios.

Findings

MCDL effectively estimates local parameters conditioned on boundary data.

MCDL generalizes to spatially invariant parameter estimation from a single configuration.

The method provides a new perspective on parameter estimation in Markov random fields.

Abstract

In this paper we discuss a method, which we call Minimum Conditional Description Length (MCDL), for estimating the parameters of a subset of sites within a Markov random field. We assume that the edges are known for the entire graph $G=(V,E)$. Then, for a subset $U\subset V$, we estimate the parameters for nodes and edges in $U$ as well as for edges incident to a node in $U$, by finding the exponential parameter for that subset that yields the best compression conditioned on the values on the boundary $\partial U$. Our estimate is derived from a temporally stationary sequence of observations on the set $U$. We discuss how this method can also be applied to estimate a spatially invariant parameter from a single configuration, and in so doing, derive the Maximum Pseudo-Likelihood (MPL) estimate.