We investigate different ways of generating approximate solutions to the inverse problem of pairwise Markov random field (MRF) model learning. We focus mainly on the inverse Ising problem, but discuss also the somewhat related inverse Gaussian problem. In both cases, the belief propagation algorithm can be used in closed form to perform inference tasks. Our approach consists in taking the Bethe mean-field solution as a reference point for further perturbation procedures. We remark indeed that both the natural gradient and the best link to be added to a maximum spanning tree (MST) solution can be computed analytically. These observations open the way to many possible algorithms, able to find approximate sparse solutions compatible with belief propagation inference procedures. Experimental tests on various datasets with refined $L_0$ or $L_1$ regularization procedures indicate that this approach may be a competitive and useful alternative to existing ones.
