reference point, based on the Plefka’s expansion and on results of Section \[sec:cumulant\]. A third method, based on strong coupling expansion and leading to a dual weight propagation algorithm (DWP) is presented in Section \[sec:dwp\]. Merits of these methods differs, which makes them complementary to each other. The first one is particularly useful when the underlying graph structure is not given; the second one, by giving explicitly the natural gradient direction, can be used to reduce the number of parameters to tune; finally the third one can be fast and exact for given very sparse structure, assuming in addition no local fields. For sake of comparison, we explain in Section \[sec:penal\] how to adapt standard $L_0$ and $L_1$ norm-penalized sparsity inducing optimization framework for finding relevant approximate solutions to the inverse GMRF problem in our context. Comparison is then made in Section \[sec:experiments\] with our own IPS based approach, in addition to other numerical experiments illustrating performances of the other methods.

Preliminaries {#sec:prelim}
=============

Inverse Ising problem
---------------------

In this section we consider binary variables ($x_i\in\{0,1\}$), which at our convenience may be also written as spin variables $s_i = 2x_i-1\in\{-1,1\}$. We assume that from a set of historical observations, the 