On conditional independence and log-convexity
Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1137-1147.

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.

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.

DOI : 10.1214/11-AIHP431
Classification : 62H05, 62M40, 62H17, 62J10, 05C50, 11C20, 15A15
Mots-clés : 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 
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Matúš, František. On conditional independence and log-convexity. Annales de l'I.H.P. Probabilités et statistiques, Tome 48 (2012) no. 4, pp. 1137-1147. doi : 10.1214/11-AIHP431. http://www.numdam.org/articles/10.1214/11-AIHP431/

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