We survey some recent work where, motivated by traffic inference, we design in parallel two concurrent models, an Ising and a Gaussian ones, with the constraint that they are suitable for ''belief-propagation'' based inference. In order to build these model, we study how a Bethe mean-field solution to inverse problems obtained with a maximum spanning tree of pairwise mutual information, can serve as a reference point for further perturbation procedures. We consider three different ways along this idea: the first one is based on an explicit natural gradient formula; the second one is a link by link construction based on iterative proportional scaling; the last one relies on a duality transformation leading to a loop correction propagation algorithm on a dual factor graph.