Groups acting on Gaussian graphical models

TL;DR

This paper explores the symmetry groups acting on Gaussian graphical models, revealing their structure, computing orbit dimensions, and analyzing implications for invariant estimators and robustness.

Contribution

It explicitly characterizes the maximal matrix groups acting on concentration matrices in Gaussian graphical models, linking group structure to graph combinatorics and estimator properties.

Findings

Dimension of orbit space depends on graph combinatorics

Characterization of models as transformation families when dimension is zero

Lower bounds on sample size for equivariant estimators and robustness analysis

Abstract

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.