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B0879
**Title: **Bayesian model selection in the space of Gaussian models invariant by permutation symmetry
**Authors: **Bartosz Kolodziejek - Politechnika Warszawska (Poland) **[presenting]**

**Abstract: **Multivariate centered Gaussian models for the random variable $X = (X_1,\ldots, X_p)$, invariant under the action of a subgroup of the group of permutations on $\{1,\ldots, p\}$ are considered. Using the representation 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 method with several examples. Further, we present novel results on the normalizing constant of Diaconis-Ylvisaker conjugate prior when $X$ satisfies conditional independencies described by a decomposable graph and the permutation group describing symmetries is a subgroup of the automorphism group of the graph.