  Pairwise models like the Ising model or the generalized Potts model have
found many successful applications in fields like physics, biology, and
economics. Closely connected is the problem of inverse statistical mechanics,
where the goal is to infer the parameters of such models given observed data.
An open problem in this field is the question of how to train these models in
the case where the data contain additional higher-order interactions that are
not present in the pairwise model. In this work, we propose an approach based
on Energy-Based Models and pseudolikelihood maximization to address these
complications: we show that hybrid models, which combine a pairwise model and a
neural network, can lead to significant improvements in the reconstruction of
pairwise interactions. We show these improvements to hold consistently when
compared to a standard approach using only the pairwise model and to an
approach using only a neural network. This is in line with the general idea
that simple interpretable models and complex black-box models are not
necessarily a dichotomy: interpolating these two classes of models can allow to
keep some advantages of both.
