Loop corrections in spin models through density consistency

TL;DR

This paper introduces a new computational method for approximating marginals in Markov random fields that incorporates loop corrections, significantly improving accuracy over traditional methods like Bethe-Peierls and cluster variational approaches.

Contribution

The proposed density consistency scheme extends belief propagation by including cycle correlations, offering more accurate marginal estimates in complex graphical models.

Findings

Significantly improves marginal estimates over Bethe-Peierls approximation.

Exact up to the $d^{-4}$ order for the critical inverse temperature expansion.

Outperforms the plaquette cluster variational method in many cases.

Abstract

Computing marginal distributions of discrete or semidiscrete Markov random fields (MRFs) is a fundamental, generally intractable problem with a vast number of applications in virtually all fields of science. We present a new family of computational schemes to approximately calculate the marginals of discrete MRFs. This method shares some desirable properties with belief propagation, in particular, providing exact marginals on acyclic graphs, but it differs with the latter in that 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 the adaptive Thouless-Anderson-Palmer method, but it differs with the latter in that 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. The results on finite-dimensional Isinglike models show a significant improvement with respect to the Bethe-Peierls (tree) approximation in all cases and with respect to the plaquette cluster variational method approximation in many cases. In particular, for the critical inverse temperature $\beta_{c}$ of the homogeneous hypercubic lattice, the expansion of $\left(d\beta_{c}\right)^{-1}$ around $d=\infty$ of the proposed scheme is exact up to the $d^{-4}$ order, whereas the two latter are exact only up to the $d^{-2}$ order.