Gérard Letac and Hélène Massam initiated in their 2007 Annals of Statistics article [1] "Wishart distributions for decomposable graphs" a systematic mathematical study of Wishart matrices subject to natural statistical constraints.

Classical Wishart matrices are supported on the cone $Sym^+(p,{\bf R})$ of symmetric positive definite matrices. In modern statistics, Wishart matrices are considered on:

- cones of matrices with obligatory zeros (sparsity)

- cones of matrices with subsets of equal terms (coloring).

These multiple-shape parameter Wishart distributions are useful in high dimensional statistical inference.

In this talk I will present some recent results obtained

in the paper [2], jointly with H. Ishi(Osaka), B. Kolodziejek(Warsaw) and H. Massam.

In [2] we consider multivariate Gaussian models $N(0,\Sigma)$ for the random variable $Z=(Z_1,\ldots, Z_p)$, invariant under the action of a subgroup $\Gamma$ of the group $\mathfrak{S}_p$ of permutations on $\{1,\ldots, p\}$.

We compute the normalization constants for Wishart laws on colored subcones of $Sym^+(p,{\bf R})$.

The statistical objective is a Bayesian model selection in the class ${\mathcal C}$ of such complete Gaussian models invariant by the action of a subgroup $\Gamma$ of the symmetric group $\mathfrak{S}_p$, also called saturated RCOP models.

Using the representation theory of the symmetric group $\mathfrak{S}_p$ 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 ${\mathcal C}$.

We illustrate our results with Frets' Heads example of dimension $4$ and a high-dimensional example in the case of cyclic groups.

[1] G. Letac, H. Massam, Wishart distributions for decomposable graphs, Ann. Statist. Volume 35, Number 3(2007), 1278-1323.

[2] P. Graczyk, H. Ishi, B. Kolodziejek, H. Massam,

Model selection in the space of Gaussian models invariant by symmetry, to appear in Annals of Statistics, 2022. arXiv:2004.03503