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We propose a novel approach to the inverse Ising problem which employs therecently introduced Density Consistency approximation (DC) to determine themodel parameters (couplings and external fields) maximizing the likelihood ofgiven empirical data. This method allows for closed-form expressions of theinferred parameters as a function of the first and second empirical moments.Such expressions have a similar structure to the small-correlation expansionderived by Sessak and Monasson, of which they provide an improvement in thecase of non-zero magnetization at low temperatures, as well as in presence ofrandom external fields. The present work provides an extensive comparison withmost common inference methods used to reconstruct the model parameters inseveral regimes, i.e. by varying both the network topology and the distributionof fields and couplings. The comparison shows that no method is uniformlybetter than every other one, but DC appears nevertheless as one of the mostaccurate and reliable approaches to infer couplings and fields from first andsecond moments in a significant range of parameters.

A Density Consistency approach to the inverse Ising problem