empirical mean $\hat m_i$ (resp. covariance $\hat \chi_{ij}$) is given for each variable $s_i$ (resp. each pair of variable $(s_i,s_j)$). In this case, from Jayne’s maximum entropy principle [@Jaynes], imposing these moments to the joint distribution leads to a model pertaining to the exponential family, the so-called Ising model: $$\label{def:ising}
{\mathcal{P}}({\bf s}) = \frac{1}{Z[{{\bf J}},{{\bf h}}]}\exp\bigl(\sum_ih_is_i+\sum_{i,j}J_{ij}s_is_j\bigr)$$ where the local fields ${{\bf h}}=\{h_i\}$ and the coupling constants ${{\bf J}}=\{J_{ij}\}$ are the Lagrange multipliers associated respectively to mean and covariance constraints when maximizing the entropy of ${\mathcal{P}}$. They are obtained as minimizers of the dual optimization problem, namely $$\label{eq:iipb}
({{{\bf h}}}^\star,{{{\bf J}}}^\star) = \operatorname*{argmin}_{({{{\bf h}}},{{{\bf J}}})} {\mathcal{L}}[{{\bf h}},{{\bf J}}],$$ with $$\label{def:LL}
{\mathcal{L}}[{{\bf h}},{{\bf J}}] = \log Z[{{\bf h}},{{\bf J}}]-\sum_ih_i{\hat m}_i-\sum_{ij}J_{ij}\hat m_{ij}$$ the log likelihood. This leads to invert the linear response equations: $$\begin{aligned}
\frac{\partial \log Z}{\partial h_i}[{{\bf h}},{{\bf J}}] &= {\hat m}_i\label{eq:cei}\\[0.2cm]
\frac{\partial \log Z}{\partial J_{ij}}[{{\bf h}},{{\bf J}}] &= \hat m_{ij},\label{eq:ccij}\end{aligned}$$ $\hat m_{ij} = {\hat m}_i{\hat m}_j+{{\hat \chi}}_{ij}$ being the empirical expectation of $s_is_j$. As noted e.g. in [@CoMo], the solution is minimizing the cross entropy, a Kullback-Leibler distance between the empirical distribution ${{\hat P}}$ based on historical data and the Ising model: $$\label{eq:dkl}
D_{KL} [\hat{\cal P}\Vert {\cal P}] = \log Z[{{\bf 