a pseudo-likelihood. This is an increasingly popular method in Bayesian statistics [@bissiri2013general] and in machine learning [@catonibook; @alquier2015properties; @begin2016pac]. We define the pseudo-posterior by $$\begin{aligned}
\tilde{\pi}_{\lambda}({\rm d}\nu)
\propto \exp\left[-\lambda \ell(\nu,\mathcal{D}) \right]
 \pi({\rm d}\nu),
 \label{pseudo_post}\end{aligned}$$ the pseudo-likelihood being $\exp\left[-\lambda \ell(\nu,\mathcal{D}) \right]$. The term $\ell(\nu,\mathcal{D}) $ can be specified by the user. Two examples are provided in Section \[MAIN\]. As a replacement of the likelihood, this term plays the role of the empirical evidence. More specially

-   the role of $\exp\left[-\lambda \ell(\nu,\mathcal{D}) \right]$ is to give more weight to the density $\nu$ when it fits the data well;

-   the role of $\pi({\rm d}\nu)$, the prior, is to restrict the posterior to the space of densities (and even give more weight to low-rank matrices if needed);

-   $\lambda>0$ is a free parameter that allows to tune the balance between evidence from the data and prior information.

We finally define the pseudo-posterior mean (also refered to as Gibbs estimator, PAC-Bayesian estimator or EWA, for exponentially weighted aggregate [@catonibook; @dalalyan2008aggregation]): $$\begin{aligned}
\tilde{\rho}_{\lambda}
= \int \nu \tilde{\pi}_{\lambda}(d\nu).\end{aligned}$$ The definition of the estimator $\tilde{\rho}_{\lambda} $ based on the pseudo-posterior $\tilde{\pi}_{\lambda}$ is actually validated by the theoretical results from Section \[MAIN\].

Definition of the prior
-----------------------

In 