Computing bounds for entropy of stationary Z^d Markov random fields

TL;DR

This paper develops efficient methods to approximate the entropy of stationary $ ext{Z}^d$-Gibbs measures satisfying strong spatial mixing, providing converging bounds with polynomial-time computability for $d=2$.

Contribution

It introduces a novel approach to compute entropy bounds for stationary $ ext{Z}^d$-Gibbs measures with strong spatial mixing, including polynomial-time algorithms for the two-dimensional case.

Findings

Sequences of upper and lower entropy bounds converge to the true entropy.

Approximations are accurate within any desired $ ext{epsilon}$ for $d=2$.

Computations are polynomial in $1/ ext{epsilon}$ for $d=2$.

Abstract

For any stationary $\mZ^d$-Gibbs measure that satisfies strong spatial mixing, we obtain sequences of upper and lower approximations that converge to its entropy. In the case, $d=2$, these approximations are efficient in the sense that the approximations are accurate to within $\epsilon$ and can be computed in time polynomial in $1/\epsilon$.