2013/08/19 by Patrick Rebeschini, Ramon van Handel · 1 citation
Mathematics · Physics and Astronomy · #Gibbs measure #Markov Chains and Monte Carlo Methods #Markov chain #Markov process #Mathematical proof #Monte Carlo method #Random Matrices and Applications #Range (aeronautics) #Statistical Mechanics and Entropy #Statistical mechanics #Uniqueness #Variable-order Markov model #math-ph #math.MP #math.PR #math.ST #stat.TH
paper · pdf · doi:10.1007/s10955-014-1087-7
published as J. Stat. Phys. 157, 234-281 (2014) · 55 pages
arxiv created 2013/08/19 · openalex publication_date 2014/08/07 · arxiv updated 2015/02/04 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
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.