We consider multivariate centered Gaussian models for the random variable $Z=(Z_1,\ldots, Z_p)$, invariant under the action of a subgroup of the group of permutations on $\{ 1,\ldots, p\}$. 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 $\Sigma$ and also the analytic expression of the normalizing constant of the Diaconis-Ylvisaker conjugate prior for the precision parameter $K=\Sigma^{ -1}$. 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 $p=100$.
Categories: math.ST math-ph math.MP math.RT stat.TH