The Annals of Statistics

Groups acting on Gaussian graphical models

Jan Draisma, Sonja Kuhnt, and Piotr Zwiernik

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Gaussian graphical models have become a well-recognized tool for the analysis of conditional independencies within a set of continuous random variables. From an inferential point of view, it is important to realize that they are composite exponential transformation families. We reveal this structure by explicitly describing, for any undirected graph, the (maximal) matrix group acting on the space of concentration matrices in the model. The continuous part of this group is captured by a poset naturally associated to the graph, while automorphisms of the graph account for the discrete part of the group. We compute the dimension of the space of orbits of this group on concentration matrices, in terms of the combinatorics of the graph; and for dimension zero we recover the characterization by Letac and Massam of models that are transformation families. Furthermore, we describe the maximal invariant of this group on the sample space, and we give a sharp lower bound on the sample size needed for the existence of equivariant estimators of the concentration matrix. Finally, we address the issue of robustness of these estimators by computing upper bounds on finite sample breakdown points.

Article information

Ann. Statist., Volume 41, Number 4 (2013), 1944-1969.

First available in Project Euclid: 23 October 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62H99: None of the above, but in this section 62F35: Robustness and adaptive procedures
Secondary: 54H15: Transformation groups and semigroups [See also 20M20, 22-XX, 57Sxx]

Gaussian graphical models covariance matrix concentration matrix robust estimator breakdown point equivariant estimator transformation families


Draisma, Jan; Kuhnt, Sonja; Zwiernik, Piotr. Groups acting on Gaussian graphical models. Ann. Statist. 41 (2013), no. 4, 1944--1969. doi:10.1214/13-AOS1130.

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  • Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis, 3rd ed. Wiley, Hoboken, NJ.
  • Andersson, S. A. and Klein, T. (2010). On Riesz and Wishart distributions associated with decomposable undirected graphs. J. Multivariate Anal. 101 789–810.
  • Andersson, S. A. and Perlman, M. D. (1993). Lattice models for conditional independence in a multivariate normal distribution. Ann. Statist. 21 1318–1358.
  • Andersson, S. A., Madigan, D., Perlman, M. D. and Triggs, C. M. (1995). On the relation between conditional independence models determined by finite distributive lattices and by directed acyclic graphs. J. Statist. Plann. Inference 48 25–46.
  • Barndorff-Nielsen, O. (1983). On a formula for the distribution of the maximum likelihood estimator. Biometrika 70 343–365.
  • Barndorff-Nielsen, O., Blæsild, P., Jensen, J. L. and Jørgensen, B. (1982). Exponential transformation models. Proc. Roy. Soc. London Ser. A 379 41–65.
  • Barrett, W. W., Johnson, C. R. and Loewy, R. (1996). The real positive definite completion problem: Cycle completability. Mem. Amer. Math. Soc. 122 viii$+$69.
  • Becker, C. (2005). Iterative proportional scaling based on a robust start estimator. In Classification—The Ubiquitous Challenge (C. Weihs and W. Gaul, eds.) 248–255. Springer, Berlin.
  • Borel, A. (1991). Linear Algebraic Groups, 2nd ed. Graduate Texts in Mathematics 126. Springer, New York.
  • Buhl, S. L. (1993). On the existence of maximum likelihood estimators for graphical Gaussian models. Scand. J. Stat. 20 263–270.
  • Davies, P. L. and Gather, U. (2005). Breakdown and groups. Ann. Statist. 33 977–1035.
  • Davies, P. L. and Gather, U. (2007). The breakdown point—Examples and counterexamples. REVSTAT 5 1–17.
  • Donoho, D. L. (1982). Breakdown properties of multivariate location estimators. Ph.D. thesis, Harvard Univ.
  • Donoho, D. and Huber, P. J. (1983). The notion of breakdown point. In A Festschrift for Erich L. Lehmann 157–184. Wadsworth, Belmont, CA.
  • Draisma, J., Kuhnt, S. and Zwiernik, P. (2013). Supplement to “Groups acting on Gaussian graphical models.” DOI:10.1214/13-AOS1130SUPP.
  • Drton, M. and Richardson, T. S. (2008). Graphical methods for efficient likelihood inference in Gaussian covariance models. J. Mach. Learn. Res. 9 893–914.
  • Eaton, M. L. (1989). Group Invariance Applications in Statistics. NSF-CBMS Regional Conference Series in Probability and Statistics, 1. IMS, Hayward, CA.
  • Finegold, M. and Drton, M. (2011). Robust graphical modeling of gene networks using classical and alternative $t$-distributions. Ann. Appl. Stat. 5 1057–1080.
  • Fisher, R. A. (1934). Two new properties of mathematical likelihood. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 144 285–307.
  • Gottard, A. and Pacillo, S. (2006). On the impact of contaminations in graphical Gaussian models. Stat. Methods Appl. 15 343–354.
  • Gottard, A. and Pacillo, S. (2010). Robust concentration graph model selection. Comput. Statist. Data Anal. 54 3070–3079.
  • Hampel, F. R. (1971). A general qualitative definition of robustness. Ann. Math. Statist. 42 1887–1896.
  • James, W. and Stein, C. (1961). Estimation with quadratic loss. In Proc. 4th Berkeley Sympos. Math. Statist. and Prob., Vol. I 361–379. Univ. California Press, Berkeley, CA.
  • Konno, Y. (2001). Inadmissibility of the maximum likelihood estimator of normal covariance matrices with the lattice conditional independence. J. Multivariate Anal. 79 33–51.
  • Kuhnt, S. and Becker, C. (2003). Sensitivity of graphical modeling against contamination. In Between Data Science and Applied Data Analysis (M. Schader, W. Gaul and M. Vichi, eds.) 279–287. Springer, Berlin.
  • Lauritzen, S. L. (1996). Graphical Models. Oxford Statistical Science Series 17. Oxford Univ. Press, New York.
  • Lehmann, E. L. and Romano, J. P. (2005). Testing Statistical Hypotheses, 3rd ed. Springer, New York.
  • Letac, G. and Massam, H. (2007). Wishart distributions for decomposable graphs. Ann. Statist. 35 1278–1323.
  • Lopuhaä, H. P. and Rousseeuw, P. J. (1991). Breakdown points of affine equivariant estimators of multivariate location and covariance matrices. Ann. Statist. 19 229–248.
  • Malyšev, F. M. (1977). Closed subsets of roots and the cohomology of regular subalgebras. Mat. Sb. 104(146) 140–150, 176.
  • Maronna, R. A., Martin, R. D. and Yohai, V. J. (2006). Robust Statistics: Theory and Methods. Wiley, Chichester.
  • Miyamura, M. and Kano, Y. (2006). Robust Gaussian graphical modeling. J. Multivariate Anal. 97 1525–1550.
  • Reid, N. (1995). The roles of conditioning in inference. Statist. Sci. 10 138–157, 173–189, 193–196.
  • Schervish, M. J. (1995). Theory of Statistics. Springer, New York.
  • Stahel, W. (1981). Robust estimation: Infinitesimal optimality and covariance matrix estimators. Ph.D. thesis, ETH, Zürich.
  • Sun, D. and Sun, X. (2005). Estimation of the multivariate normal precision and covariance matrices in a star-shape model. Ann. Inst. Statist. Math. 57 455–484.
  • Uhler, C. (2012). Geometry of maximum likelihood estimation in Gaussian graphical models. Ann. Statist. 40 238–261.
  • Vogel, D. and Fried, R. (2011). Elliptical graphical modelling. Biometrika 98 935–951.

Supplemental materials

  • Supplementary material: Proofs and more on the structure of $\mathbf{P}_\mathcal{C}$. We provide the proof of Proposition 6.1 and more results on the structure of the poset $\mathbf{P}_\mathcal{C}$ that link our work to Andersson and Perlman (1993).