Graphical Gaussian models associated to a homogeneous graph with permutation symmetries

TL;DR

This paper develops Bayesian model selection methods for Gaussian graphical models with permutation symmetries, providing explicit formulas for priors and demonstrating the approach with a small example.

Contribution

It derives an explicit normalizing constant for the conjugate prior and applies Bayesian model selection to permutation-invariant Gaussian models.

Findings

Explicit normalizing constant for the conjugate prior derived.

Bayesian model selection framework established for symmetric Gaussian models.

Illustrative example with a 5-dimensional model provided.

Abstract

We consider multivariate centered Gaussian models for the random vector $(Z^1,\ldots, Z^p)$, whose conditional structure is described by a homogeneous graph and which is invariant under the action of a permutation subgroup. The following paper concerns with model selection within colored graphical Gaussian models, when the underlying conditional dependency graph is known. We derive an analytic expression of the normalizing constant of the Diaconis-Ylvisaker conjugate prior for the precision parameter and perform Bayesian model selection in the class of graphical Gaussian models invariant by the action of a permutation subgroup. We illustrate our results with a toy example of dimension $5$.