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

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.

DOI : https://doi.org/10.1214/11-AIHP431
Classification:  62H05,  62M40,  62H17,  62J10,  05C50,  11C20,  15A15
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
@article{AIHPB_2012__48_4_1137_0,
     author = {Mat\'u\v s, Franti\v sek},
     title = {On conditional independence and log-convexity},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {48},
     number = {4},
     year = {2012},
     pages = {1137-1147},
     doi = {10.1214/11-AIHP431},
     zbl = {1253.62036},
     mrnumber = {3052406},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2012__48_4_1137_0}
}
Matúš, František. On conditional independence and log-convexity. Annales de l'I.H.P. Probabilités et statistiques, Volume 48 (2012) no. 4, pp. 1137-1147. doi : 10.1214/11-AIHP431. http://www.numdam.org/item/AIHPB_2012__48_4_1137_0/

[1] S. Amari. Differential-Geometrical Methods in Statistics. Lecture Notes in Statistics 28. Springer, Berlin, 1985. | MR 788689 | Zbl 0559.62001

[2] M. B. Averintsev. On a method of describing random fields with discrete argument. Problemy Peredachi Informacii 6 (1970) 100-108. | MR 329038

[3] J. Besag. Spatial interaction and the statistical analysis of lattice systems. J. R. Stat. Soc. Ser. B Stat. Methodol. 36 (1974) 192-236. | MR 373208 | Zbl 0327.60067

[4] G. Birkhoff. Lattice Theory. AMS Colloquium Publications XXV. AMS, Providence, RI, 1967. | MR 227053 | Zbl 0153.02501

[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. | MR 205315 | Zbl 0129.10703

[6] N. N. Chentsov. Statistical Decision Rules and Optimal Inference. AMS, Providence, RI, 1982. Translated from Russian, Nauka, Moscow, 1972. | MR 343398 | Zbl 0484.62008

[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. | MR 1064553 | Zbl 0736.62081

[8] I. Csiszár and F. Matúš. Generalized maximum likelihood estimates for exponential families. Probab. Theory Related Fields 141 (2008) 213-246. | MR 2372970 | Zbl 1133.62039

[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. | MR 568718 | Zbl 0444.62064

[10] J. N. Darroch and T. P. Speed. Additive and multiplicative models and interactions. Ann. Statist. 11 (1983) 724-738. | MR 707924 | Zbl 0556.62032

[11] A. P. Dawid. Conditional independence in statistical theory (with discussion). J. R. Stat. Soc. Ser. B Stat. Methodol. 41 (1979) 1-31. | MR 535541 | Zbl 0408.62004

[12] M. Drton, B. Sturmfels and S. Sullivant. Lectures on Algebraic Statistics. Birkhäuser, Basel, 2009. | MR 2723140 | Zbl 1166.13001

[13] M. Drton and H. Xiao. Smoothness of Gaussian conditional independence models. Contemp. Math. 516 (2010) 155-177. | MR 2730747 | Zbl 1196.62055

[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. | MR 329039 | Zbl 0261.60043

[16] G. R. Grimmett. Probability on Graphs. Cambridge Univ. Press, Cambridge, 2010. | MR 2723356 | Zbl 1228.60003

[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. | JFM 45.0123.01

[19] J. G. Kemeny, J. L. Snell and A. W. Knapp. Denumerable Markov Chains. Springer, New York, 1976. | MR 407981 | Zbl 0348.60090

[20] S. L. Lauritzen. Graphical Models. Oxford Univ. Press, Oxford, 1996. | MR 1419991 | Zbl 0907.62001

[21] R. Lněnička and F. Matúš. On Gaussian conditional independence structures. Kybernetika (Prague) 43 (2007) 327-342. | MR 2362722 | Zbl 1144.60302

[22] F. Matúš. On equivalence of Markov properties over undirected graphs. J. Appl. Probab. 29 (1992) 745-749. | MR 1174448 | Zbl 0753.68080

[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. | Zbl 0764.60002

[24] F. Matúš. Conditional independences among four random variables III: Final conclusion. Combin. Probab. Comput. 8 (1999) 269-276. | MR 1702569 | Zbl 0941.60004

[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. | Zbl 1111.68685

[26] J. Moussouris. Gibbs and Markov random systems with constraints. J. Stat. Phys. 10 (1974) 11-33. | MR 432132

[27] C. J. Preston. Generalized Gibbs states and Markov random fields. Adv. in Appl. Probab. 5 (1973) 242-261. | MR 405645 | Zbl 0318.60085

[28] S. Sherman. Markov random fields and Gibbs random fields. Israel J. Math. 14 (1973) 92-103. | MR 321185 | Zbl 0255.60047

[29] T. P. Speed and H. T. Kiiveri. Gaussian Markov distributions over finite graphs. Ann. Statist. 14 (1986) 138-150. | MR 829559 | Zbl 0589.62033

[30] F. Spitzer. Markov random fields and Gibbs ensembles. Amer. Math. Monthly 78 (1971) 142-154. | MR 277036 | Zbl 0241.60094

[31] M. Studený. Probabilistic Conditional Independence Structures. Springer, New York, 2005. | Zbl 1070.62001