Bethe-Peierls approximation and the inverse Ising problem

We apply the Bethe-Peierls approximation to the inverse Ising problem and show how the linear response relation leads to a simple method for reconstructing couplings and fields of the Ising model. This reconstruction is exact on tree graphs, yet its computational expense is comparable to those of other mean-field methods. We compare the performance of this method to the independent-pair, naive mean-field, and Thouless-Anderson-Palmer approximations, the Sessak-Monasson expansion, and susceptibility propagation on the Cayley tree, SK model and random graph with fixed connectivity. At low temperatures, Bethe reconstruction outperforms all of these methods, while at high temperatures it is comparable to the best method available so far ( the Sessak-Monasson method). The relationship between Bethe reconstruction and other mean-field methods is discussed.