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Journal of Statistical Physics

, Volume 157, Issue 2, pp 234–281 | Cite as

Comparison Theorems for Gibbs Measures

  • Patrick Rebeschini
  • Ramon van HandelEmail author
Article

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.

Keywords

Dobrushin comparison theorem Gibbs measures Filtering algorithms 

Notes

Acknowledgments

This work was supported in part by NSF Grants DMS-1005575 and CAREER-DMS-1148711, and by the ARO through PECASE award W911NF-14-1-0094.

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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Sherrerd Hall, Princeton UniversityPrincetonUSA

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