Comparison Theorems for Gibbs Measures

TL;DR

This paper extends the classical Dobrushin comparison theorem for Gibbs measures, allowing broader applicability by using influence-based criteria and analyzing associated Markov chains, with applications to high-dimensional filtering algorithms.

Contribution

It develops generalized Dobrushin comparison theorems based on influences between blocks, significantly expanding their applicability beyond the classical uniqueness criterion.

Findings

Extended comparison theorems applicable to a wider class of models.

Provided detailed analysis of Markov chains related to the theorems.

Applied results to high-dimensional sequential Monte Carlo algorithms.

Abstract

The Dobrushin comparison theorem is a powerful tool to bound the difference between the marginals of high-dimensional probability distributions in terms of their local specifications. Originally introduced to prove uniqueness and decay of correlations of Gibbs measures, it has been widely used in statistical mechanics as well as in the analysis of algorithms on random fields and interacting Markov chains. However, the classical comparison theorem requires validity of the Dobrushin uniqueness criterion, essentially restricting its applicability in most models to a small subset of the natural parameter space. In this paper we develop generalized Dobrushin comparison theorems in terms of influences between blocks of sites, in the spirit of Dobrushin-Shlosman and Weitz, that substantially extend the range of applicability of the classical comparison theorem. Our proofs are based on the analysis of an associated family of Markov chains. We develop in detail an application of our main results to the analysis of sequential Monte Carlo algorithms for filtering in high dimension.