Model selection in the space of Gaussian models invariant by symmetry

Abstract

We consider multivariate centered Gaussian models for the random variable $\mathit{Z}=({\mathit{Z}}_1,\dots ,{\mathit{Z}}_{\mathit{p}})$, invariant under the action of a subgroup of the group of permutations on $\{1,\dots ,\mathit{p}\}$. Using the representation theory of the symmetric group on the field of reals, we derive the distribution of the maximum likelihood estimate of the covariance parameter Σ and also the analytic expression of the normalizing constant of the Diaconis–Ylvisaker conjugate prior for the precision parameter $\mathit{K}={\mathrm{\Sigma }}^{-1}$. We can thus perform Bayesian model selection in the class of complete Gaussian models invariant by the action of a subgroup of the symmetric group, which we could also call complete RCOP models. We illustrate our results with a toy example of dimension 4 and several examples for selection within cyclic groups, including a high- dimensional example with $\mathit{p}=100$.