Electronic Journal of Probability

Phase transitions in nonlinear filtering

Patrick Rebeschini and Ramon van Handel

Full-text: Open access


It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information.  While the existing theory provides a rather complete picture of classical filtering models, many infinite dimensional problems are outside its scope.  Far from being a technical issue, the infinite dimensional setting gives rise to surprising phenomena and new questions in filtering theory. The aim of this paper is to discuss some elementary examples, conjectures, and general theory that arise in this setting, and to highlight connections with problems in statistical mechanics and ergodic theory.  In particular, we exhibit a simple example of a uniformly ergodic model in which ergodicity of the filter undergoes a phase transition, and we develop some qualitative understanding as to when such phenomena can and cannot occur.We also discuss closely related problems in the setting of conditional Markov random fields.

Article information

Electron. J. Probab. Volume 20 (2015), paper no. 7, 46 pp.

Accepted: 1 February 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37A50: Relations with probability theory and stochastic processes [See also 60Fxx and 60G10]
Secondary: 60G35: Signal detection and filtering [See also 62M20, 93E10, 93E11, 94Axx] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B26: Phase transitions (general) 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.)

filtering in infinite dimension conditional ergodicity and mixing phase transitions

This work is licensed under a Creative Commons Attribution 3.0 License.


Rebeschini, Patrick; van Handel, Ramon. Phase transitions in nonlinear filtering. Electron. J. Probab. 20 (2015), paper no. 7, 46 pp. doi:10.1214/EJP.v20-3281. https://projecteuclid.org/euclid.ejp/1465067113

Export citation


  • Blackwell, David. The entropy of functions of finite-state Markov chains. 1957 Transactions of the first Prague conference on information theory, Statistical decision functions, random processes held at Liblice near Prague from November 28 to 30, 1956 pp. 13–20 Publishing House of the Czechoslovak Academy of Sciences, Prague.
  • Blackwell, David; Dubins, Lester. Merging of opinions with increasing information. Ann. Math. Statist. 33 1962 882–886.
  • Bleher, P. M.; Ruiz, J.; Zagrebnov, V. A. On the purity of the limiting Gibbs state for the Ising model on the Bethe lattice. J. Statist. Phys. 79 (1995), no. 1-2, 473–482.
  • Bleher, P. M.; Ruiz, J.; Zagrebnov, V. A. On the phase diagram of the random field Ising model on the Bethe lattice. J. Statist. Phys. 93 (1998), no. 1-2, 33–78.
  • Bovier, Anton. Statistical mechanics of disordered systems. A mathematical perspective. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 2006. xiv+312 pp. ISBN: 978-0-521-84991-3; 0-521-84991-8.
  • Carlson, J. M.; Chayes, J. T.; Chayes, L.; Sethna, J. P.; Thouless, D. J. Bethe lattice spin glass: the effects of a ferromagnetic bias and external fields. I. Bifurcation analysis. J. Statist. Phys. 61 (1990), no. 5-6, 987–1067.
  • Carlson, J. M.; Chayes, J. T.; Sethna, J. P.; Thouless, D. J. Bethe lattice spin glass: the effects of a ferromagnetic bias and external fields. II. Magnetized spin-glass phase and the de Almeida-Thouless line. J. Statist. Phys. 61 (1990), no. 5-6, 1069–1084.
  • Chayes, J. T.; Chayes, L.; Sethna, James P.; Thouless, D. J. A mean field spin glass with short-range interactions. Comm. Math. Phys. 106 (1986), no. 1, 41–89.
  • Chigansky, Pavel; van Handel, Ramon. A complete solution to Blackwell's unique ergodicity problem for hidden Markov chains. Ann. Appl. Probab. 20 (2010), no. 6, 2318–2345.
  • Conze, J. P. Entropie d'un groupe abïélien de transformations. (French) Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 25 (1972/73), 11–30.
  • Coquille, Loren; Velenik, Yvan. A finite-volume version of Aizenman-Higuchi theorem for the 2d Ising model. Probab. Theory Related Fields 153 (2012), no. 1-2, 25–44.
  • Cover, Thomas M.; Thomas, Joy A. Elements of information theory. Second edition. Wiley-Interscience [John Wiley & Sons], Hoboken, NJ, 2006. xxiv+748 pp. ISBN: 978-0-471-24195-9; 0-471-24195-4.
  • Crisan, D.; Rozovskiĭ, B. Introduction. The Oxford handbook of nonlinear filtering, 1–15, Oxford Univ. Press, Oxford, 2011.
  • Deuschel, Jean-Dominique; Stroock, Daniel W. Large deviations. Pure and Applied Mathematics, 137. Academic Press, Inc., Boston, MA, 1989. xiv+307 pp. ISBN: 0-12-213150-9.
  • Douc, Randal; Gassiat, Elisabeth; Landelle, Benoit; Moulines, Eric. Forgetting of the initial distribution for nonergodic hidden Markov chains. Ann. Appl. Probab. 20 (2010), no. 5, 1638–1662.
  • Evans, William; Kenyon, Claire; Peres, Yuval; Schulman, Leonard J. Broadcasting on trees and the Ising model. Ann. Appl. Probab. 10 (2000), no. 2, 410–433.
  • Fernïández, R.; Pfister, C.-E. Global specifications and nonquasilocality of projections of Gibbs measures. Ann. Probab. 25 (1997), no. 3, 1284–1315.
  • Föllmer, Hans. On entropy and information gain in random fields. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 26 (1973), 207–217.
  • Föllmer, On the global Markov property, Quantum fields–-algebras, processes (Proc. Sympos., Univ. Bielefeld, Bielefeld, 1978), Springer, Vienna, 1980, pp. 293–302.
  • Föllmer, Hans. Random fields and diffusion processes. École d'Été de Probabilités de Saint-Flour XV–XVII, 1985–87, 101–203, Lecture Notes in Math., 1362, Springer, Berlin, 1988.
  • Frigessi, Arnoldo; Martinelli, Fabio; Stander, Julian. Computational complexity of Markov chain Monte Carlo methods for finite Markov random fields. Biometrika 84 (1997), no. 1, 1–18.
  • Georgii, Hans-Otto. Gibbs measures and phase transitions. Second edition. de Gruyter Studies in Mathematics, 9. Walter de Gruyter & Co., Berlin, 2011. xiv+545 pp. ISBN: 978-3-11-025029-9.
  • Georgii, Hans-Otto; Häggström, Olle; Maes, Christian. The random geometry of equilibrium phases. Phase transitions and critical phenomena, Vol. 18, 1–142, Phase Transit. Crit. Phenom., 18, Academic Press, San Diego, CA, 2001.
  • Glasner, Eli. Ergodic theory via joinings. Mathematical Surveys and Monographs, 101. American Mathematical Society, Providence, RI, 2003. xii+384 pp. ISBN: 0-8218-3372-3.
  • Goldstein, S.; Kuik, R.; Schlijper, A. G. Entropy and global Markov properties. Comm. Math. Phys. 126 (1990), no. 3, 469–482.
  • Goldstein, Sheldon. Remarks on the global Markov property. Comm. Math. Phys. 74 (1980), no. 3, 223–234.
  • Goldstein, Sheldon; Kuik, Roelof; Lebowitz, Joel L.; Maes, Christian. From PCAs to equilibrium systems and back. Comm. Math. Phys. 125 (1989), no. 1, 71–79.
  • Grimmett, Geoffrey. The random-cluster model. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 333. Springer-Verlag, Berlin, 2006. xiv+377 pp. ISBN: 978-3-540-32890-2; 3-540-32890-4
  • Kunita, Hiroshi. Asymptotic behavior of the nonlinear filtering errors of Markov processes. J. Multivariate Anal. 1 (1971), 365–393.
  • Lebowitz, Joel L.; Maes, Christian; Speer, Eugene R. Statistical mechanics of probabilistic cellular automata. J. Statist. Phys. 59 (1990), no. 1-2, 117–170.
  • Liggett, Thomas M. Interacting particle systems. Reprint of the 1985 original. Classics in Mathematics. Springer-Verlag, Berlin, 2005. xvi+496 pp. ISBN: 3-540-22617-6
  • Liggett, Thomas M. Conditional association and spin systems. ALEA Lat. Am. J. Probab. Math. Stat. 1 (2006), 1–19.
  • Louis, Pierre-Yves. Ergodicity of PCA: equivalence between spatial and temporal mixing conditions. Electron. Comm. Probab. 9 (2004), 119–131 (electronic).
  • J. C. Mattingly, phErgodicity of dissipative SPDEs, Lecture notes, École d'été de Probabilités de Saint-Flour, July 8–21, 2007 (unpublished), 2007.
  • Meyn, S. P.; Tweedie, R. L. Markov chains and stochastic stability. Communications and Control Engineering Series. Springer-Verlag London, Ltd., London, 1993. xvi+ 548 pp. ISBN: 3-540-19832-6
  • Mézard, Marc; Montanari, Andrea. Information, physics, and computation. Oxford Graduate Texts. Oxford University Press, Oxford, 2009. xiv+569 pp. ISBN: 978-0-19-857083-7
  • Abbe, Emmanuel; Montanari, Andrea. Conditional random fields, planted constraint satisfaction and entropy concentration. Approximation, randomization, and combinatorial optimization, 332–346, Lecture Notes in Comput. Sci., 8096, Springer, Heidelberg, 2013.
  • Elchanan Mossel, Joe Neeman, and Allan Sly, phReconstruction and estimation in the planted partition model, Probab. Th. Rel. Fields (2014), to appear.
  • Nishimori, Hidetoshi. Duality in finite-dimensional spin glasses. J. Stat. Phys. 126 (2007), no. 4-5, 977–986.
  • Rebeschini, Patrick; van Handel, Ramon. Comparison theorems for Gibbs measures. J. Stat. Phys. 157 (2014), no. 2, 234–281.
  • Rebeschini, Patrick; van Handel, Ramon. Comparison theorems for Gibbs measures. J. Stat. Phys. 157 (2014), no. 2, 234–281.
  • Daniel Revuz and Marc Yor, phContinuous martingales and Brownian motion, third ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. (2000h:60050)
  • Stettner, Łukasz. On invariant measures of filtering processes. Stochastic differential systems (Bad Honnef, 1988), 279–292, Lecture Notes in Control and Inform. Sci., 126, Springer, Berlin, 1989.
  • Stuart, A. M. Inverse problems: a Bayesian perspective. Acta Numer. 19 (2010), 451–559.
  • Tong, Xin Thomson; van Handel, Ramon. Ergodicity and stability of the conditional distributions of nondegenerate Markov chains. Ann. Appl. Probab. 22 (2012), no. 4, 1495–1540.
  • Tong, Xin Thomson; van Handel, Ramon. Conditional ergodicity in infinite dimension. Ann. Probab. 42 (2014), no. 6, 2243–2313.
  • van Enter, Aernout C. D.; Fernéndez, Roberto; Sokal, Alan D. Regularity properties and pathologies of position-space renormalization-group transformations: scope and limitations of Gibbsian theory. J. Statist. Phys. 72 (1993), no. 5-6, 879–1167.
  • Van Handel, Ramon. Discrete time nonlinear filters with informative observations are stable. Electron. Commun. Probab. 13 (2008), 562–575.
  • van Handel, Ramon. Observability and nonlinear filtering. Probab. Theory Related Fields 145 (2009), no. 1-2, 35–74.
  • van Handel, Ramon. The stability of conditional Markov processes and Markov chains in random environments. Ann. Probab. 37 (2009), no. 5, 1876–1925.
  • van Handel, Ramon. Uniform observability of hidden Markov models and filter stability for unstable signals. Ann. Appl. Probab. 19 (2009), no. 3, 1172–1199.
  • van Handel, Ramon. On the exchange of intersection and supremum of $\sigma$-fields in filtering theory. Israel J. Math. 192 (2012), no. 2, 763–784.
  • von Weizsäcker, Heinrich. Exchanging the order of taking suprema and countable intersections of $\sigma $-algebras. Ann. Inst. H. Poincar� Sect. B (N.S.) 19 (1983), no. 1, 91–100.
  • Winkler, Gerhard. Image analysis, random fields and Markov chain Monte Carlo methods. A mathematical introduction. Second edition. With 1 CD-ROM (Windows). Applications of Mathematics (New York), 27. Stochastic Modelling and Applied Probability. Springer-Verlag, Berlin, 2003. xvi+387 pp. ISBN: 3-540-44213-8.