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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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).