A new class of colored Gaussian graphical models with explicit normalizing constants
Adam Chojecki, Piotr Graczyk, Hideyuki Ishi, Bartosz Kołodziejek
TL;DR
The paper tackles the computational bottleneck of evaluating Diaconis–Ylvisaker normalizing constants $I_{g,c}(oldsymbol ext{delta},D)$ in Bayesian model selection for colored Gaussian graphical models. It introduces Color Elimination-Regular (CER) graphs and Block-Cholesky (BC) and Diagonally Commutative BC (DCBC) spaces to obtain explicit gamma-type integral formulas in terms of a finite set of structure constants, with an efficient Random Method plus Gram–Cholesky procedure for their computation. The authors show CER graphs extend decomposable RCOP models, connect the single-color case to Bose–Mesner algebras, and relate the framework to RDAG models, thereby broadening principled Bayesian structure learning for CGGMs in high dimensions. This work thus advances tractable Bayesian inference for CGGMs by linking combinatorial graph structure with algebraic spaces that admit closed-form normalizing constants, enabling scalable high-dimensional applications.
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.