Lattices of Graphical Gaussian Models with Symmetries

TL;DR

This paper explores lattice structures of graphical Gaussian models with symmetries, providing explicit search algorithms and identifying model classes suitable for efficient model selection in high-dimensional settings.

Contribution

It identifies four lattice-structured model classes with symmetries, and develops explicit algorithms for model search within these classes.

Findings

Four model classes form complete lattices for model inclusion.

Two classes are most suitable for model search.

An explicit search algorithm is provided for one class.

Abstract

In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, model classes which place equality restrictions on concentrations or partial correlations have previously been introduced in the literature. The models can be represented by vertex and edge coloured graphs. The need for model selection methods makes it imperative to understand the structure of model classes. We identify four model classes that form complete lattices of models with respect to model inclusion, which qualifies them for an Edwards-Havr\'anek model selection procedure. Two classes turn out most suitable for a corresponding model search. We obtain an explicit search algorithm for one of them and provide a model search example for the other.