The randomization by Wishart laws and the Fisher information

Consider the centered Gaussian vector $X$ in $\R^n$ with covariance matrix $\Sigma.$ Randomize $\Sigma$ such that $\Sigma^{-1}$ has a Wishart distribution with shape parameter $p>(n-1)/2$ and mean $p\sigma.$ We compute the density $f_{p,\sigma}$ of $X$ as well as the Fisher information $I_p(\sigma)$ of the model $(f_{p,\sigma} )$ when $\sigma$ is the parameter. For using the Cram\'er-Rao inequality, we also compute the inverse of $I_p(\sigma)$. The important point of this note is the fact that this inverse is a linear combination of two simple operators on the space of symmetric matrices, namely $\P(\sigma)(s)=\sigma s \sigma$ and $(\sigma\otimes \sigma)(s)=\sigma \, \mathrm{trace}(\sigma s)$. The Fisher information itself is a linear combination $\P(\sigma^{-1})$ and $\sigma^{-1}\otimes \sigma^{-1}.$ Finally, by randomizing $\sigma$ itself, we make explicit the minoration of the second moments of an estimator of $\sigma$ by the Van Trees inequality: here again, linear combinations of $\P(u)$ and $u\otimes u$ appear in the results.