The randomization by Wishart laws and the Fisher information
Consider the centered Gaussian vector $X$ in $\R^n$ with covariance matrix $ Σ.$ Randomize $Σ$ such that $ Σ^{-1}$ has a Wishart distribution with shape parameter $p>(n-1)/2$ and mean $pσ.$ We compute the density $f_{p,σ}$ of $X$ as well as the Fisher information $I_p(σ)$ of the model $(f_{p,σ} )$ when $σ$ is the parameter. For using the Cramér-Rao inequality, we also compute the inverse of $I_p(σ)$. 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 $¶(σ)(s)=σs σ$ and $(σ\otimes σ)(s)=σ\, \mathrm{trace}(σs)$. The Fisher information itself is a linear combination $¶(σ^{-1})$ and $σ^{-1}\otimes σ^{-1}.$ Finally, by randomizing $σ$ itself, we make explicit the minoration of the second moments of an estimator of $σ$ by the Van Trees inequality: here again, linear combinations of $¶(u)$ and $u\otimes u$ appear in the results.