incomplete measurement case. We consider the (incomplete) pseudo-likelihood as $$\begin{aligned}
 \ell^{prob-incomplete}(\nu,\mathcal{D}) = \sum_{\mathbf{a}\in \mathcal{A}}
\sum_{\mathbf{s}\in \mathcal{R}^n}
\left[{\rm Tr}(\nu P_\mathbf{s}^\mathbf{a})
-\hat{p}_{\mathbf{a},\mathbf{s}}\right]^2,\end{aligned}$$ where $ \mathcal{A} \varsubsetneq \mathcal{E}^n $. The study in this case will be the object of future works.

Open questions include faster algorithms based on optimization (in the spirit of [@alquier2015properties]). Also, from a theoretical perspective, the most important question is the minimax lower bound.

Acknowledgements {#acknowledgements .unnumbered}
================

Both authors gratefully acknowledge financial support from GENES and by the French National Research Agency (ANR) under the grant Labex Ecodec (ANR-11- LABEX-0047). P.Alquier gratefully acknowledges financial support from the research programme [*New Challenges for New Data*]{} from LCL and GENES, hosted by the [*Fondation du Risque*]{}.

Proofs {#Appendix}
======

We first remind here a version of Hoeffding’s inequality for bounded random variables.

\[bernstein ine\] Let $Y_i, i = 1,\ldots,n$ be $n$ independent random variables with $|Y_i|\leq b$ a.s., and $\mathbb{E}(Y_i)=0$. Then, for any $\lambda>0$, $$\mathbb{E}\exp\left(\frac{\lambda}{n}
 \sum_{i=1}^{n} Y_i\right)
        \leq
 \exp\left(\frac{\lambda^2 b^2}{8n} \right) .$$

Preliminary lemmas for the proof of Theorem \[thrm: rate 1\]
------------------------------------------------------------

\[lem app bern 1\] For any $\lambda>0$, we have $$\mathbb{E}\exp\left(\lambda \left<p_\nu - p^0,
 p^0-\hat{p}\right>_F\right)
  \leq \exp\left[ \frac{\lambda^2}{4m}\|p^0
  -p_\nu\|_F^2  \right],$$ $$\mathbb{E}\exp\left(-\lambda \left<p_\nu - p^0,
p^0-\hat{p}\right>_F\right)
  \leq
  \exp\left[ \frac{\lambda^2}{4m}\|p^0
  