to look at least for a good trade-off between likelihood and sparsity.

Concerning the Inverse Ising Problem (IIP), the existing approaches fall mainly in the following categories:

-   Purely computational efficient approaches rely on various optimization schemes of the log likelihood [@LeGaKo] or on pseudo-likelihood [@HoTi] along with sparsity constraints to select the only relevant features.

-   Common analytical approaches are based on the Plefka expansion [@Plefka] of the Gibbs free energy by making the assumption that the coupling constants $J_{ij}$ are small. The picture is then of a weakly correlated uni-modal probability measure. For example, the recent approach proposed in [@CoMo] is based on this assumption.

-   Another possibility is to assume that relevant coupling $J_{ij}$ have locally a tree-like structure. The Bethe approximation [@YeFrWe] can then be used with possibly loop corrections. Again this corresponds to having a weakly correlated uni-modal probability measure and these kind of approaches are referred to as pseudo-moment matching methods in the literature for the reason explained in the previous section. For example the approaches proposed in [@KaRo; @WeTe; @MoMe; @YaTa] are based on this assumption.

-   In the case where a multi-modal distribution is expected, then a model with many attraction basins 