Model selection in the space of Gaussian models invariant by symmetry
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We consider multivariate centred Gaussian models for the random variable Z=(Z_1,..., Z_p), 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.
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