Loop corrections in spin models through density consistency

Computing marginal distributions of discrete or semi-discrete Markov Random Fields (MRF) is a fundamental, generally intractable, problem with a vast number of applications on virtually all fields of science. We present a new family of computational schemes to calculate approximately marginals of discrete MRFs. This method shares some desirable properties with Belief Propagation, in particular providing exact marginals on acyclic graphs; but at difference with it, it includes some loop corrections, i.e. it takes into account correlations coming from all cycles in the factor graph. It is also similar to Adaptive TAP, but at difference with it, the consistency is not on the first two moments of the distribution but rather on the value of its density on a subset of values. Results on random connectivity and finite dimensional Ising and Edward-Anderson models show a significant improvement with respect to the Bethe-Peierls (tree) approximation in all cases, and significant improvement with respect to Cluster Variational Method and Loop Corrected Bethe approximation in many cases. In particular, for the critical inverse temperature $\beta_{c}$ of the homogeneous hypercubic lattice, the $1/d$ expansion of $\left(d\beta_{c}\right)^{-1}$ of the proposed scheme is exact up to the $d^{-3}$ order, whereas the two latter are exact only up to the $d^{-2}$ order.