# Wishart laws on sparse and colored matrix cones with applications to statistics

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

- cones of matrices with obligatory zeros (sparsity)

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

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

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

In this talk I will present some recent results, strongly inspired by the article [1] and obtained jointly with H. Ishi(Osaka), B. Kolodziejek(Warsaw), S. Mamane(Johannesburg), H. Massam(Toronto) and H. Ochiai(Kyushu).

1. Letac--Massam Conjecture.

In [2], we proved, for graphical models with nearest neighbour interactions, a conjecture on parameter range stated by Letac and Massam in the article [1]. Our result has statistical application of full description of well-defined Bayes estimators and priors in Bayesian analysis of models with nearest neighbour interactions.

2. Normalization constants for Wishart laws on colored subcones of $Sym^+(n,{\bf R})$. In [3] 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 $\{1,\ldots, p\}$.

The statistical objective is a 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, S. Mamane, H. Ochiai, On the Letac-Massam conjecture on cones $Q_{A_n}$, Proceedings of the Japan Academy, Series A, Mathematical Sciences 93/3(2017), 16-21.

[3] P. Graczyk, H. Ishi, B. Kolodziejek, H. Massam, Model selection in the space of Gaussian models invariant by symmetry, preprint (2020)

[4]P. Graczyk, H. Ishi, Riesz measures and Wishart laws associated to quadratic maps, Journal of Jap.Math.Soc.66(2014), 317-348.

[5] P. Graczyk, H. Ishi, B. Kolodziejek, Wishart laws and variance function on homogeneous cones, Prob. Mathematical Stat. Vol. 39, Fasc. 2 (2019), pp. 337-360.

[6] P. Graczyk, H. Ishi, S. Mamane, Wishart exponential families on cones related to tridiagonal matrices, Annals of the Institute of Statistical Mathematics 71(2019), pp. 439–471