The Bethe approximation for solving the inverse Ising problem: a comparison with other inference methods

The inverse Ising problem consists in inferring the coupling constants of an Ising model given the correlation matrix. The fastest methods for solving this problem are based on mean-field approximations, but which one performs better in the general case is still not completely clear. In the first part of this work, I summarize the formulas for several mean-field approximations and I derive new analytical expressions for the Bethe approximation, which allow one to solve the inverse Ising problem without running the susceptibility propagation algorithm (thus avoiding the lack of convergence). In the second part, I compare the accuracy of different mean-field approximations on several models (diluted ferromagnets and spin glasses) defined on random graphs and regular lattices, showing which one is in general more effective. A simple improvement over these approximations is proposed. Also a fundamental limitation is found in using methods based on TAP and Bethe approximations in the presence of an external field.