We consider multivariate centred Gaussian models for the random variable
Z=(Z1,…,Zp), invariant under the action of a subgroup of the group of
permutations on {1,…,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 K=Σ−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.Comment: 34 pages, 4 figures, 5 table