A new class of colored Gaussian graphical models with explicit normalizing constants

TL;DR

This paper introduces a new class of colored Gaussian graphical models with explicit normalizing constants, enabling more efficient Bayesian model selection for models with symmetry and sparsity constraints.

Contribution

The authors develop a tractable framework for a broad family of CGGMs by identifying conditions for closed-form normalizing constants and characterizing the graph structures that induce them.

Findings

Explicit formulas for normalizing constants in new subclasses of models.

Characterization of graph classes that guarantee tractability.

Enhanced Bayesian structure learning in high-dimensional settings.

Abstract

We study Bayesian model selection in colored Gaussian graphical models (CGGMs), which combine sparsity of conditional independencies with symmetry constraints encoded by vertex- and edge-colored graphs. A computational bottleneck in Bayesian inference for CGGMs is the evaluation of Diaconis-Ylvisaker normalizing constants, given by gamma-type integrals over cones of precision matrices with prescribed zeros and equality constraints. While explicit formulas are known for standard Gaussian graphical models only in special cases (e.g. decomposable graphs) and for a limited class of RCOP models, no general tractable framework has been available for broader families of CGGMs. We introduce a new subclass of RCON models for which these normalizing constants admit closed-form expressions. On the algebraic side, we identify conditions on spaces of colored precision matrices that guarantee tractability of the associated integrals, leading to Block-Cholesky spaces (BC-spaces) and Diagonally Commutative Block-Cholesky spaces (DCBC-spaces). On the combinatorial side, we characterize the colored graphs inducing such spaces via a color perfect elimination ordering and a 2-path regularity condition, and define the resulting Color Elimination-Regular (CER) graphs and their symmetric variants. This class strictly extends decomposable graphs in the uncolored setting and contains all RCOP models associated with decomposable graphs. In the one-color case, our framework reveals a close connection between DCBC-spaces and Bose-Mesner algebras. For models defined on BC- and DCBC-spaces, we derive explicit closed-form formulas for the normalizing constants in terms of a finite collection of structure constants and propose an efficient method for computing them in the commutative case. Our results broaden the range of CGGMs amenable to principled Bayesian structure learning in high-dimensional applications.