V_1,\ldots, V_d & \sim \text{ i.i.d}  \text{ uniform distribution on
 the unit sphere,} \\
 (\gamma_1, \ldots,\gamma_d) & \sim \mathcal{D}ir(\alpha_1,\dots,\alpha_d),
 \\
 \rho & = \sum_{i=1}^d \gamma_i V_i V_i^{\dagger}\end{aligned}$$ where $\mathcal{D}ir(\alpha_1,\dots,\alpha_d)$ is the Dirichlet distribution with parameters $\alpha_1,\dots,\alpha_d>0$.

To get an approximate rank-$1$ matrix $\rho$, one can take all parameters of the Dirichlet distribution equal to a constant that is very closed to 0 (e.g $ \alpha_1=\ldots=\alpha_d=\frac{1}{d} $). And a typical drawing will lead to one of the $\gamma_i's $ close to $1 $ and the others close to $0 $. See [@wallach2009rethinking] for more discussion on choosing the parameters for Dirichlet distribution. Theoretical recommendations for the $ \alpha_i $’s are given in Section \[MAIN\] below.

We could impose the $V_i$’s to be orthogonal in practice. The theoretical results would be unchanged, however, the implementation of our method would become trickier. Note that to sample from the uniform distribution on the sphere is rather easy. We can for example simulate $\tilde{V}_i$ from any isotropic distribution, e.g. $\mathcal{N}(0,\mathbb{I})$ and define $V_i := \tilde{V}_i / \|\tilde{V}_i\|$.

PAC-Bayesian estimation and analysis {#MAIN}
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Pseudo-likelihoods
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Here, we consider two natural ways to compare a theoretical density $\rho $ and the observations: first $p_{\mathbf{a},\mathbf{s}} $ should 