of learned graph structures. To complete this overview, let us mention also that some authors proposed information based structure learning methods [@NeBaSaSh] quite in line with some approaches to be discussed in the present paper.

In some preceding work dealing with a road traffic inference application, with large scale and real time specifications [@FuLaFo; @FuLaAu; @FuHaLa], we have noticed that these methods could not be used blindly without drastic adjustment, in particular to be compatible with belief propagation. This led us to develop some heuristic models related to the Bethe approximation. The present work is an attempt to give a theoretical basis and firmer ground to these heuristics and to develop new ones.

The paper is organized as follows: in Section \[sec:prelim\] we review some standard statistical physics approaches to the IIP, mainly based on perturbation expansions, and derive some new and useful expressions of susceptibility coefficients in Section \[sec:cumulant\]. In Section \[sec:iip\] we explain what we mean by the *Bethe reference point* and develop an iterative proportional scaling (IPS) based method to incrementally, link by link, refine this approximate solution, both for the inverse Ising and inverse GMRF problems. Section \[sec:perturbation\] explores a second way to refine the Bethe 