Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On conditional independence and log-convexity

František Matúš

Full-text: Open access

Abstract

If conditional independence constraints define a family of positive distributions that is log-convex then this family turns out to be a Markov model over an undirected graph. This is proved for the distributions on products of finite sets and for the regular Gaussian ones. As a consequence, the assertion known as Brook factorization theorem, Hammersley–Clifford theorem or Gibbs–Markov equivalence is obtained.

Résumé

Si des contraintes d’indépendance conditionnelle définissent une famille de distributions positives qui est log-convexe, alors cette famille doit être un modèle de Markov sur un graphe non-dirigé. Ceci est démontré pour les distributions sur le produits d’ensembles finis et pour les distributions gaussiennes régulières. Par conséquent, l’assertion connue comme le théorème de factorisation de Brook, le théorème de Hammersley–Clifford ou l’équivalence de Gibbs–Markov est obtenue.

Article information

Source
Ann. Inst. H. Poincaré Probab. Statist., Volume 48, Number 4 (2012), 1137-1147.

Dates
First available in Project Euclid: 16 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1353098443

Digital Object Identifier
doi:10.1214/11-AIHP431

Mathematical Reviews number (MathSciNet)
MR3052406

Zentralblatt MATH identifier
1253.62036

Subjects
Primary: 62H05: Characterization and structure theory 62M40: Random fields; image analysis
Secondary: 62H17: Contingency tables 62J10: Analysis of variance and covariance 05C50: Graphs and linear algebra (matrices, eigenvalues, etc.) 11C20: Matrices, determinants [See also 15B36] 15A15: Determinants, permanents, other special matrix functions [See also 19B10, 19B14]

Keywords
Conditional independence Markov properties Factorizable distributions Graphical Markov models Log-convexity Gibbs–Markov equivalence Markov fields Hammersley–Clifford theorem Contingency tables Gibbs potentials Multivariate Gaussian distributions Positive definite matrices Covariance selection model

Citation

Matúš, František. On conditional independence and log-convexity. Ann. Inst. H. Poincaré Probab. Statist. 48 (2012), no. 4, 1137--1147. doi:10.1214/11-AIHP431. https://projecteuclid.org/euclid.aihp/1353098443


Export citation

References

  • [1] S. Amari. Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics 28. Springer, Berlin, 1985.
  • [2] M. B. Averintsev. On a method of describing random fields with discrete argument. Problemy Peredachi Informacii 6 (1970) 100–108.
  • [3] J. Besag. Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B Stat. Methodol. 36 (1974) 192–236.
  • [4] G. Birkhoff. Lattice Theory. AMS Colloquium Publications XXV. AMS, Providence, RI, 1967.
  • [5] D. Brook. On the distinction between the conditional probability and joint probability approaches in the specification of nearest-neighbour systems. Biometrika 51 (1964) 481–483.
  • [6] N. N. Chentsov. Statistical Decision Rules and Optimal Inference. AMS, Providence, RI, 1982. Translated from Russian, Nauka, Moscow, 1972.
  • [7] P. Clifford. Markov random fields in statistics. In: Disorder in Physical Systems: A Volume in Honour of J. M. Hammersley 19–32. G. Grimmett and D. Welsh (Eds). Oxford University Press, New York, 1990.
  • [8] I. Csiszár and F. Matúš. Generalized maximum likelihood estimates for exponential families. Probab. Theory Related Fields 141 (2008) 213–246.
  • [9] J. N. Darroch, S. L. Lauritzen and T. P. Speed. Markov fields and log-linear interaction models for contingency tables. Ann. Statist. 8 (1980) 522–539.
  • [10] J. N. Darroch and T. P. Speed. Additive and multiplicative models and interactions. Ann. Statist. 11 (1983) 724–738.
  • [11] A. P. Dawid. Conditional independence in statistical theory (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 41 (1979) 1–31.
  • [12] M. Drton, B. Sturmfels and S. Sullivant. Lectures on Algebraic Statistics. Birkhäuser, Basel, 2009.
  • [13] M. Drton and H. Xiao. Smoothness of Gaussian conditional independence models. Contemp. Math. 516 (2010) 155–177.
  • [14] D. Griffeath. Markov and Gibbs fields with finite state space on graphs. Unpublished manuscript, 1973.
  • [15] G. R. Grimmett. A theorem about random fields. Bull. Lond. Math. Soc. 5 (1973) 81–84.
  • [16] G. R. Grimmett. Probability on Graphs. Cambridge Univ. Press, Cambridge, 2010.
  • [17] J. M. Hammersley and P. E. Clifford. Markov fields on finite graphs and lattices. Unpublished manuscript, 1971.
  • [18] F. Hausdorff. Grundzüge der Mengenlehre. Veit, Leipzig, 1914.
  • [19] J. G. Kemeny, J. L. Snell and A. W. Knapp. Denumerable Markov Chains. Springer, New York, 1976.
  • [20] S. L. Lauritzen. Graphical Models. Oxford Univ. Press, Oxford, 1996.
  • [21] R. Lněnička and F. Matúš. On Gaussian conditional independence structures. Kybernetika (Prague) 43 (2007) 327–342.
  • [22] F. Matúš. On equivalence of Markov properties over undirected graphs. J. Appl. Probab. 29 (1992) 745–749.
  • [23] F. Matúš. Ascending and descending conditional independence relations. In: Transactions of the 11-th Prague Conference on Information Theory, Statistical Decision Functions and Random Processes B 189–200. Academia, Prague, 1992.
  • [24] F. Matúš. Conditional independences among four random variables III: Final conclusion. Combin. Probab. Comput. 8 (1999) 269–276.
  • [25] F. Matúš. Conditional independences in Gaussian vectors and rings of polynomials. In Proceedings of WCII 2002. LNAI 3301 152–161. G. Kern-Isberner, W. Rödder and F. Kulmann (Eds). Springer, Berlin, 2005.
  • [26] J. Moussouris. Gibbs and Markov random systems with constraints. J. Stat. Phys. 10 (1974) 11–33.
  • [27] C. J. Preston. Generalized Gibbs states and Markov random fields. Adv. in Appl. Probab. 5 (1973) 242–261.
  • [28] S. Sherman. Markov random fields and Gibbs random fields. Israel J. Math. 14 (1973) 92–103.
  • [29] T. P. Speed and H. T. Kiiveri. Gaussian Markov distributions over finite graphs. Ann. Statist. 14 (1986) 138–150.
  • [30] F. Spitzer. Markov random fields and Gibbs ensembles. Amer. Math. Monthly 78 (1971) 142–154.
  • [31] M. Studený. Probabilistic Conditional Independence Structures. Springer, New York, 2005.