Skip to main content
SpringerLink
Log in

Partially Ordered Models

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We provide a formal definition and study the basic properties of partially ordered random fields (PORF). These systems were proposed to model textures in image processing and to represent independence relations between random variables in statistics (in the latter case they are known as Bayesian networks). Our random fields are a generalization of probabilistic cellular automata (PCA) and their theory has features intermediate between that of discrete-time processes and the theory of statistical mechanical lattice fields. Its proper definition is based on the notion of partially ordered specification (POS), in close analogy to the theory of Gibbs measures. This paper contains two types of results. First, we present the basic elements of the general theory of PORFs: basic geometrical issues, definition in terms of conditional probability kernels, extremal decomposition, extremality and triviality, reconstruction starting from single-site kernels, relations between POM and Gibbs fields. Second, we prove three uniqueness criteria that correspond to the criteria known as uniform boundedness, Dobrushin uniqueness and disagreement percolation in the theory of Gibbs measures.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Belitsky, V., Logan Ritchie, T.: Improved lower bounds for the critical probability of oriented bond percolation in two dimensions. J. Stat. Phys. 122(2), 279–302 (2006)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  2. Cressie, N., Davidson, J.L.: Image analysis with partially ordered Markov models. Comput. Stat. Data Anal. 29(1), 1–26 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Cressie, N., Davidson, J.L., Hua, X.: Texture synthesis and pattern recognition for partially ordered Markov models. Pattern Recogn., 32 (1999)

  4. Deveaux, V.: Modèles markoviens partiellement orientés. Approche géométrique des Automates cellulaires probabilistes. PhD thesis, Université de Rouen, France (2008) (in English). http://tel.archives-ouvertes.fr/tel-00325051

  5. Deveaux, V.: Geometrical approach of probabilistic cellular automata (2010, in preparation)

  6. Dobrushin, R.L., Kolafa, J., Shlosman, S.B.: Phase diagram of the two-dimensional Ising antiferromagnet. Computer-assisted proof. Commun. Math. Phys. 102(1), 89–103 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  7. Dachian, S., Nahapetian, B.S.: Description of random fields by means of one-point conditional distributions and some applications. Markov Process. Relat. Fields 7(2), 193–214 (2001)

    MATH  MathSciNet  Google Scholar 

  8. Dachian, S., Nahapetian, B.S.: Description of specifications by means of probability distributions in small volumes under condition of very weak positivity. J. Stat. Phys. 117(1–2), 281–300 (2004)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  9. Dachian, S., Nahapetian, B.S.: On Gibbsianness of random fields. Markov Process. Relat. Fields 15, 81–104 (2009)

    MATH  MathSciNet  Google Scholar 

  10. Dobrushin, R.L.: Perturbation methods of the theory of Gibbsian fields. In: Lectures on Probability Theory and Statistics, Saint-Flour, 1994. Lecture Notes in Math., vol. 1648, pp. 1–66. Springer, Berlin (1996)

    Chapter  Google Scholar 

  11. Ferrari, P.A., Galves, A.: Construction of stochastic processes, coupling and regeneration. XIII Escuela Venezolana de Matemática

  12. Fernández, R., Maillard, G.: Chains with complete connections and one-dimensional Gibbs measures. Electron. J. Probab. 9(6), 145–176 (2004) (electronic)

    MathSciNet  Google Scholar 

  13. Fernández, R., Maillard, G.: Chains with complete connections: general theory, uniqueness, loss of memory and mixing properties. J. Stat. Phys. 118(3–4), 555–588 (2005)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  14. Fernández, R., Maillard, G.: Construction of a specification from its singleton part. ALEA Lat. Am. J. Probab. Math. Stat. 2, 297–315 (2006) (electronic)

    MATH  MathSciNet  Google Scholar 

  15. Georgii, H.-O.: Gibbs Measures and Phase Transitions. de Gruyter Studies in Mathematics, vol. 9. Gruyter, Berlin (1988)

    MATH  Google Scholar 

  16. Georgii, H.-O., Häggström, O., Maes, C.: The random geometry of equilibrium phases. In: Phase Transitions and Critical Phenomena. Phase Transit. Crit. Phenom., vol. 18, pp. 1–142. Academic Press, San Diego (2001)

    Chapter  Google Scholar 

  17. Goldstein, S., Kuik, R., Lebowitz, J.L., Maes, C.: From PCAs to equilibrium systems and back. Commun. Math. Phys. 125(1), 71–79 (1989)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  18. Jensen, F.V., Nielsen, T.D.: Bayesian Networks and Decision Graphs. Information Science and Statistics, 2nd edn. Springer, New York (2007)

    MATH  Google Scholar 

  19. Lebowitz, J.L., Martin-Löf, A.: On the uniqueness of the equilibrium state for Ising spin systems. Commun. Math. Phys. 25, 276–282 (1972)

    Article  ADS  Google Scholar 

  20. Lebowitz, J.L., Maes, C., Speer, E.R.: Probabilistic cellular automata: some statistical mechanical considerations. In: 1989 Lectures in Complex Systems, Santa Fe, NM, 1989. Santa Fe Inst. Stud. Sci. Complexity Lectures, vol. II, pp. 401–414. Addison-Wesley, Redwood City (1990)

    Google Scholar 

  21. Lebowitz, J.L., Maes, C., Speer, E.R.: Statistical mechanics of probabilistic cellular automata. J. Stat. Phys. 59(1–2), 117–170 (1990)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  22. Maillard, G.: Chaînes à liaisons complètes et mesures de Gibbs unidimensionnelles. PhD thesis, Université de Rouen, France, 2003. http://tel.archives-ouvertes.fr/tel-00005285/fr/

  23. Maes, C., Shlosman, S.B.: Ergodicity of probabilistic cellular automata: a constructive criterion. Commun. Math. Phys. 135, 233–251 (1991)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  24. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)

    MATH  Google Scholar 

  25. Toom, A.: Contornos, conjuntos convexos e autômatos celulares. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, 2001. 23° Colóquio Brasileiro de Matemática. [23rd Brazilian Mathematics Colloquium]

  26. van den Berg, J.: A uniqueness condition for Gibbs measures, with application to the 2-dimensional Ising antiferromagnet. Commun. Math. Phys. 152(1), 161–166 (1993)

    Article  MATH  ADS  Google Scholar 

  27. van den Berg, J., Maes, C.: Disagreement percolation in the study of Markov fields. Ann. Probab. 22(2), 749–763 (1994)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Author notes

  1. Roberto Fernández

    Present address: Department of Mathematics, Utrecht University, P.O. Box 80010, 3508, TA, Utrecht, The Netherlands

Authors and Affiliations

  1. Laboratoire de Mathématiques Raphaël Salem, UMR 6085 CNRS-Université de Rouen, Avenue de l’Université, BP.12, 76801, Saint Etienne du Rouvray, France

    Vincent Deveaux & Roberto Fernández

Authors
  1. Vincent Deveaux
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roberto Fernández
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vincent Deveaux.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deveaux, V., Fernández, R. Partially Ordered Models. J Stat Phys 141, 476–516 (2010). https://doi.org/10.1007/s10955-010-0063-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-010-0063-0

Keywords

Search

Navigation