Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

On conditional independence and log-convexity

František Matúš

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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.


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.

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

First available in Project Euclid: 16 November 2012

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Zentralblatt MATH identifier

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]

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


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.

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