Bethe-Peierls approximation and the inverse Ising model

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