Automorphism groups of Gaussian chain graph models

TL;DR

This paper characterizes the automorphism groups of Gaussian chain graph models, extending previous work to more general models, and provides efficient methods for their computation and statistical applications.

Contribution

It introduces a comprehensive analysis of automorphism groups for Gaussian chain graph models and offers efficient computational techniques using imsets.

Findings

Derived simple conditions for vanishing sub-minors of the concentration matrix.

Provided methods for efficient computation of automorphism groups without essential graph construction.

Applied results to statistical tasks like equivariant estimation and robustness.

Abstract

In this paper we extend earlier work on groups acting on Gaussian graphical models to Gaussian Bayesian networks and more general Gaussian models defined by chain graphs. We discuss the maximal group which leaves a given model invariant and provide basic statistical applications of this result. This includes equivariant estimation, maximal invariants and robustness. The computation of the group requires finding the essential graph. However, by applying Studeny's theory of imsets we show that computations for DAGs can be performed efficiently without building the essential graph. In our proof we derive simple necessary and sufficient conditions on vanishing sub-minors of the concentration matrix in the model.