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November 2012 On conditional independence and log-convexity
František Matúš
Ann. Inst. H. Poincaré Probab. Statist. 48(4): 1137-1147 (November 2012). DOI: 10.1214/11-AIHP431


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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František Matúš. "On conditional independence and log-convexity." Ann. Inst. H. Poincaré Probab. Statist. 48 (4) 1137 - 1147, November 2012.


Published: November 2012
First available in Project Euclid: 16 November 2012

zbMATH: 1253.62036
MathSciNet: MR3052406
Digital Object Identifier: 10.1214/11-AIHP431

Primary: 62H05 , 62M40
Secondary: 05C50 , 11C20 , 15A15 , 62H17 , 62J10

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

Rights: Copyright © 2012 Institut Henri Poincaré


Vol.48 • No. 4 • November 2012
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