joint distribution (\[eq:ansatz\]) is assumed with a complete graph. There is then by construction a belief propagation fixed point for which the beliefs satisfy all the constraints. This solution to be meaningful requires small correlations, so that the belief propagation fixed point be stable and unique, allowing the corresponding log likelihood to be well approximated. Otherwise, this solution is not satisfactory, but a pruning procedure, which amounts to select a sub-graph based on mutual information, can be used. The first step is to find the maximum spanning tree (MST) with mutual information taken as edges weights. How to add new links to this baseline solution in a consistent way will be the subject of the next section.

-   the indirect way consists in first inverting the potentially non-sparse correlation matrix. If the underlying interaction matrix is actually a tree, this will be visible in the inverse correlation matrix, indicated directly by the non-zero entries. Corresponding couplings are then determined through (\[eq:invchis\]). This procedure seems to work better than the previous one also when no sparsity but weak coupling is assumed. It corresponds in fact to the equations solved iteratively by the susceptibility propagation algorithm [@MoMe].

Let us first justify the 