Journal of Statistical Physics

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

Comparison Theorems for Gibbs Measures

  • Patrick Rebeschini
  • Ramon van HandelEmail author


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.


Dobrushin comparison theorem Gibbs measures Filtering algorithms 



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.


  1. 1.
    Cappé, O., Moulines, E., Rydén, T.: Inference in Hidden Markov Models. Springer Series in Statistics. Springer, New York (2005)Google Scholar
  2. 2.
    Dai Pra, P., Scoppola, B., Scoppola, E.: Sampling from a Gibbs measure with pair interaction by means of PCA. J. Stat. Phys. 149(4), 722–737 (2012)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Del Moral, P., Guionnet, A.: On the stability of interacting processes with applications to filtering and genetic algorithms. Ann. Inst. H. Poincaré Probab. Stat. 37(2), 155–194 (2001)CrossRefzbMATHGoogle Scholar
  4. 4.
    Dobrushin, R.L., Shlosman, S.B.: Constructive criterion for the uniqueness of Gibbs field. In: Statistical physics and dynamical systems (Köszeg, 1984), Program Physics, vol. 10, pp. 347–370. Birkhäuser Boston, Boston (1985)Google Scholar
  5. 5.
    Dobrušin, R.L.: Definition of a system of random variables by means of conditional distributions. Teor. Verojatnost. i Primenen. 15, 469–497 (1970)MathSciNetGoogle Scholar
  6. 6.
    Dudley, R.M.: Real Analysis and Probability, Cambridge Studies in Advanced Mathematics, vol. 74. Cambridge University Press, Cambridge (2002) (Revised reprint of the 1989 original)Google Scholar
  7. 7.
    Dyer, M., Goldberg, L.A., Jerrum, M.: Dobrushin conditions and systematic scan. Comb. Probab. Comput. 17(6), 761–779 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Dyer, M., Goldberg, L.A., Jerrum, M.: Matrix norms and rapid mixing for spin systems. Ann. Appl. Probab. 19(1), 71–107 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Föllmer, H.: Tail structure of Markov chains on infinite product spaces. Z. Wahrsch. Verw. Gebiete 50(3), 273–285 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Föllmer, H.: A covariance estimate for Gibbs measures. J. Funct. Anal. 46(3), 387–395 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Föllmer, H.: Random fields and diffusion processes. In: École d’Été de Probabilités de Saint-Flour XV–XVII, 1985–1987, Lecture Notes in Mathematics, vol. 1362, pp. 101–203. Springer, Berlin (1988)Google Scholar
  12. 12.
    Georgii, H.O.: Measures and Phase Transitions, de Gruyter Studies in Mathematics, vol. 9, 2nd edn. Walter de Gruyter & Co., Berlin (2011)CrossRefGoogle Scholar
  13. 13.
    Guionnet, A., Zegarlinski, B.: Lectures on Logarithmic Sobolev Inequalities. In: Séminaire de Probabilités, XXXVI, Lecture Notes in Mathematics, vol. 1801, pp. 1–134. Springer, Berlin (2003)Google Scholar
  14. 14.
    Kallenberg, O.: Foundations of Modern Probability. Probability and its Applications (New York), 2nd edn. Springer-Verlag, New York (2002)Google Scholar
  15. 15.
    Külske, C.: Concentration inequalities for functions of Gibbs fields with application to diffraction and random Gibbs measures. Commun. Math. Phys. 239(1–2), 29–51 (2003)ADSCrossRefzbMATHGoogle Scholar
  16. 16.
    Lebowitz, J.L., Maes, C., Speer, E.R.: Statistical mechanics of probabilistic cellular automata. J. Stat. Phys. 59(1–2), 117–170 (1990)ADSCrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Liggett, T.M.: Interacting Particle Systems. Classics in Mathematics. Springer-Verlag, Berlin (2005) (Reprint of the 1985 original)Google Scholar
  18. 18.
    Rebeschini, P., van Handel, R.: Can Local Particle Filters Beat the Curse of Dimensionality? Preprint arxiv:1301.6585 (2013)
  19. 19.
    Royden, H.L.: Real Analysis, 3rd edn. Macmillan Publishing Company, New York (1988)zbMATHGoogle Scholar
  20. 20.
    Rue, T.D.L., Fernández, R., Sokal, A.D.: How to clean a dirty floor: probabilistic potential theory and the Dobrushin uniqueness theorem. Markov Process Relat. Fields 14(1), 1–78 (2008)zbMATHGoogle Scholar
  21. 21.
    Simon, B.: The Statistical Mechanics of Lattice Gases. Princeton Series in Physics, vol. I. Princeton University Press, Princeton (1993)Google Scholar
  22. 22.
    Tatikonda, S.C.: Convergence of the sum-product algorithm. In: Information Theory Workshop, 2003. Proceedings. 2003 IEEE, pp. 222–225 (2003)Google Scholar
  23. 23.
    van der Vaart, A.W., Wellner, J.A.: Weak Convergence and Empirical Processes. Springer Series in Statistics. With Applications to Statistics. Springer-Verlag, New York (1996)Google Scholar
  24. 24.
    Villani, C.: Optimal Transport, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 338. Springer-Verlag, Berlin (2009). Old and newGoogle Scholar
  25. 25.
    Weitz, D.: Combinatorial criteria for uniqueness of Gibbs measures. Random Struct. Algorithms 27(4), 445–475 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Wu, L.: Poincaré and transportation inequalities for Gibbs measures under the Dobrushin uniqueness condition. Ann. Probab. 34(5), 1960–1989 (2006)CrossRefMathSciNetGoogle Scholar
  27. 27.
    Younes, L.: Parametric inference for imperfectly observed Gibbsian fields. Probab. Theory Relat. Fields 82(4), 625–645 (1989)CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Sherrerd Hall, Princeton UniversityPrincetonUSA

Personalised recommendations