Bethe–Peierls approximation and the inverse Ising problem

Abstract

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