The Annals of Statistics

Total positivity in Markov structures

Shaun Fallat, Steffen Lauritzen, Kayvan Sadeghi, Caroline Uhler, Nanny Wermuth, and Piotr Zwiernik

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We discuss properties of distributions that are multivariate totally positive of order two ($\mathrm{MTP}_{2}$) related to conditional independence. In particular, we show that any independence model generated by an $\mathrm{MTP}_{2}$ distribution is a compositional semi-graphoid which is upward-stable and singleton-transitive. In addition, we prove that any $\mathrm{MTP}_{2}$ distribution satisfying an appropriate support condition is faithful to its concentration graph. Finally, we analyze factorization properties of $\mathrm{MTP}_{2}$ distributions and discuss ways of constructing $\mathrm{MTP}_{2}$ distributions; in particular, we give conditions on the log-linear parameters of a discrete distribution which ensure $\mathrm{MTP}_{2}$ and characterize conditional Gaussian distributions which satisfy $\mathrm{MTP}_{2}$.

Article information

Ann. Statist., Volume 45, Number 3 (2017), 1152-1184.

Received: October 2015
Revised: May 2016
First available in Project Euclid: 13 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60E15: Inequalities; stochastic orderings 62H99: None of the above, but in this section
Secondary: 15B48: Positive matrices and their generalizations; cones of matrices

Association concentration graph conditional Gaussian distribution faithfulness graphical models log-linear interactions Markov property positive dependence


Fallat, Shaun; Lauritzen, Steffen; Sadeghi, Kayvan; Uhler, Caroline; Wermuth, Nanny; Zwiernik, Piotr. Total positivity in Markov structures. Ann. Statist. 45 (2017), no. 3, 1152--1184. doi:10.1214/16-AOS1478.

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  • [1] Ahlswede, R. and Daykin, D. E. (1978). An inequality for the weights of two families of sets, their unions and intersections. Z. Wahrsch. Verw. Gebiete 43 183–185.
  • [2] Allman, E. S., Rhodes, J. A., Sturmfels, B. and Zwiernik, P. (2015). Tensors of nonnegative rank two. Linear Algebra Appl. 473 37–53.
  • [3] Bartolucci, F. and Besag, J. (2002). A recursive algorithm for Markov random fields. Biometrika 89 724–730.
  • [4] Bartolucci, F. and Forcina, A. (2000). A likelihood ratio test for $\mathrm{MTP}_{2}$ within binary variables. Ann. Statist. 28 1206–1218.
  • [5] Bølviken, E. (1982). Probability inequalities for the multivariate normal with nonnegative partial correlations. Scand. J. Stat. 9 49–58.
  • [6] Chvátal, V. and Wu, B. (2011). On Reichenbach’s causal betweenness. Erkenntnis 76 41–48.
  • [7] Colangelo, A., Scarsini, M. and Shaked, M. (2005). Some notions of multivariate positive dependence. Insurance Math. Econom. 37 13–26.
  • [8] Dawid, A. P. (1979). Conditional independence in statistical theory. J. R. Stat. Soc. Ser. B. Stat. Methodol. 41 1–31.
  • [9] Dawid, A. P. (1980). Conditional independence for statistical operations. Ann. Statist. 8 598–617.
  • [10] Dawid, A. P. and Lauritzen, S. L. (1993). Hyper-Markov laws in the statistical analysis of decomposable graphical models. Ann. Statist. 21 1272–1317.
  • [11] Djolonga, J. and Krause, A. (2015). Scalable variational inference in log-supermodular models. In Proceedings of the International Conference on Machine Learning (ICML). Available at ArXiv:1502.06531.
  • [12] Edwards, D. (2000). Introduction to Graphical Modelling, 2nd ed. Springer, New York.
  • [13] Esary, J. D., Proschan, F. and Walkup, D. W. (1967). Association of random variables, with applications. Ann. Math. Stat. 38 1466–1474.
  • [14] Fortuin, C. M., Kasteleyn, P. W. and Ginibre, J. (1971). Correlation inequalities on some partially ordered sets. Comm. Math. Phys. 22 89–103.
  • [15] Højsgaard, S. and Lauritzen, S. L. (2008). Graphical Gaussian models with edge and vertex symmetries. J. R. Stat. Soc. Ser. B. Stat. Methodol. 70 1005–1027.
  • [16] Holland, P. W. and Rosenbaum, P. R. (1986). Conditional association and unidimensionality in monotone latent variable models. Ann. Statist. 14 1523–1543.
  • [17] Joag-Dev, K. (1983). Independence via uncorrelatedness under certain dependence structures. Ann. Probab. 11 1037–1041.
  • [18] Joe, H. (2006). Generating random correlation matrices based on partial correlations. J. Multivariate Anal. 97 2177–2189.
  • [19] Jones, B. and West, M. (2005). Covariance decomposition in undirected Gaussian graphical models. Biometrika 92 779–786.
  • [20] Karlin, S. and Rinott, Y. (1980). Classes of orderings of measures and related correlation inequalities. I. Multivariate totally positive distributions. J. Multivariate Anal. 10 467–498.
  • [21] Karlin, S. and Rinott, Y. (1983). $M$-matrices as covariance matrices of multinormal distributions. Linear Algebra Appl. 52/53 419–438.
  • [22] Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Clarendon, Oxford.
  • [23] Lebowitz, J. L. (1972). Bounds on the correlations and analyticity properties of ferromagnetic Ising spin systems. Comm. Math. Phys. 28 313–321.
  • [24] Lehmann, E. L. (1966). Some concepts of dependence. Ann. Math. Stat. 37 1137–1153.
  • [25] Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Analysis. Academic Press, London.
  • [26] Müller, A. and Scarsini, M. (2005). Archimedean copulae and positive dependence. J. Multivariate Anal. 93 434–445.
  • [27] Newman, C. M. (1983). A general central limit theorem for FKG systems. Comm. Math. Phys. 91 75–80.
  • [28] Newman, C. M. (1984). Asymptotic independence and limit theorems for positively and negatively dependent random variables. In Inequalities in Statistics and Probability (Lincoln, Neb., 1982). Institute of Mathematical Statistics Lecture Notes—Monograph Series 5 127–140. IMS, Hayward, CA.
  • [29] Ostrowski, A. (1937). Über die determinanten mit überwiegender Hauptdiagonale. Comment. Math. Helv. 10 69–96.
  • [30] Pearl, J. (1988). Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, San Mateo, CA.
  • [31] Pearl, J. and Paz, A. (1987). Graphoids: A graph based logic for reasoning about relevancy relations. In Advances in Artificial Intelligence, Vol. II (B. D. Boulay, D. Hogg and L. Steel, eds.) 357–363. North-Holland, Amsterdam.
  • [32] Peters, J. (2015). On the intersection property of conditional independence and its application to causal discovery. J. Causal Inference 3 97–108.
  • [33] Propp, J. G. and Wilson, D. B. (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures Algorithms 9 223–252.
  • [34] Reichenbach, H. (1956). The Direction of Time. Univ. California Press, Berkeley, CA.
  • [35] Royston, P., Altman, D. G. and Sauerbrei, W. (2006). Dichotomizing continuous predictors in multiple regression: A bad idea. Stat. Med. 25 127–141.
  • [36] Sadeghi, K. and Lauritzen, S. (2014). Markov properties for mixed graphs. Bernoulli 20 676–696.
  • [37] San Martín, E., Mouchart, M. and Rolin, J.-M. (2005). Ignorable common information, null sets and Basu’s first theorem. Sankhyā 67 674–698.
  • [38] Sarkar, T. K. (1969). Some lower bounds of reliability. Tech. Report, No. 124, Dept. Operations Research and Dept. Statistics, Stanford Univ., Stanford, CA.
  • [39] Slawski, M. and Hein, M. (2015). Estimation of positive definite $M$-matrices and structure learning for attractive Gaussian Markov random fields. Linear Algebra Appl. 473 145–179.
  • [40] Steel, M. and Faller, B. (2009). Markovian log-supermodularity, and its applications in phylogenetics. Appl. Math. Lett. 22 1141–1144.
  • [41] Štembera, Z., Znamenáček, K. and Poláček, K. (2012). High Risk Pregnancy and Child. Springer, Berlin.
  • [42] Studený, M. (2005). Probabilistic Conditional Independence Structures. Springer, London.
  • [43] Wermuth, N. (2012). Traceable regressions. Int. Stat. Rev. 80 415–438.
  • [44] Wermuth, N. (2015). Graphical Markov models, unifying results and their interpretation. In Wiley Statsref: Statistics Reference Online. Available at ArXiv:1505.02456.
  • [45] Wermuth, N. and Cox, D. R. (1998). On association models defined over independence graphs. Bernoulli 4 477–495.
  • [46] Wermuth, N. and Marchetti, G. M. (2014). Star graphs induce tetrad correlations: For Gaussian as well as for binary variables. Electron. J. Stat. 8 253–273.
  • [47] Whittaker, J. (1990). Graphical Models in Applied Multivariate Statistics. Wiley, Chichester.
  • [48] Xie, X., Ma, Z. and Geng, Z. (2008). Some association measures and their collapsibility. Statist. Sinica 18 1165–1183.
  • [49] Zatoński, W., Becher, H., Lissowska, J. and Wahrendorf, J. (1991). Tobacco, alcohol, and diet in the etiology of laryngeal cancer: A population-based case-control study. Cancer Causes & Control 2 3–10.
  • [50] Zwiernik, P. (2016). Semialgebraic Statistics and Latent Tree Models. Monographs on Statistics and Applied Probability 146. Chapman & Hall/CRC, Boca Raton, FL.