be close to the empirical part $\hat{p}_{\mathbf{a},\mathbf{s}} $; second $\rho $ should be close to the least square (invert) estimator $\hat{\rho} $. As we have no reason to prefer one in advance, we define and study $2 $ estimators.

### a) Distance between the probabilities: prob-estimator {#a-distance-between-the-probabilities-prob-estimator .unnumbered}

We consider $$\ell^{prob}(\nu,\mathcal{D}) = \sum_{\mathbf{a}\in\mathcal{E}^n}
\sum_{\mathbf{s}\in \mathcal{R}^n}
\left[{\rm Tr}(\nu P_\mathbf{s}^\mathbf{a})
-\hat{p}_{\mathbf{a},\mathbf{s}}\right]^2$$ and $$\begin{aligned}
\tilde{\rho}^{prob}_{\lambda}
&= \int \nu \tilde{\pi}^{prob}_{\lambda}(d\nu),
\\
\tilde{\pi}^{prob}_{\lambda}({\rm d}\nu)
& \propto \exp\left[-\lambda \ell^{prob}(\nu,\mathcal{D})
 \right] \pi({\rm d}\nu).\end{aligned}$$ Note that if we use the shortened notation $p_\nu=[{\rm Tr}(\nu P_\mathbf{s}^
\mathbf{a})]_{\mathbf{a},\mathbf{s}}$ and $\hat{p}=[\hat{p}_{\mathbf{a},
\mathbf{s}}]_{\mathbf{a},\mathbf{s}}$ then $$\ell^{prob}(\nu,\mathcal{D}) = \| p_\nu -
\hat{p}  \|^2_F$$ (Frobenius norm). This distance quantifies how far the probabilities and the empirical frequencies in the sample are.

### b) Distance between the density matrices: dens-estimator {#b-distance-between-the-density-matrices-dens-estimator .unnumbered}

Now, let us take: $$\ell^{dens}(\nu,\mathcal{D}) =
 \|\nu-\hat{\rho}\|_F^2.$$ and $$\begin{aligned}
\tilde{\rho}^{dens}_{\lambda}
&= \int \nu \tilde{\pi}^{dens}_{\lambda}(d\nu),
\\
\tilde{\pi}^{dens}_{\lambda}({\rm d}\nu)
& \propto \exp\left[-\lambda \ell^{dens}(\nu,
\mathcal{D}) \right] \pi({\rm d}\nu).\end{aligned}$$ In another words, this estimator finds a balance between prior information and closeness to the least square estimate $\hat{\rho}$. From a computational point of view, this estimator is easier to implement than the previous estimator.

Statistical properties of the estimators
----------------------------------------

\[A1\] Fix some constants $D_1>0$ and $D_2>0$ (that 