Replicated Bethe Free Energy: a Variational Principle behind Survey Propagation

A scheme to provide various mean-field-type approximation algorithms is presented by employing the Bethe free energy formalism to a family of replicated systems in conjunction with analytical continuation with respect to the number of replicas. In the scheme, survey propagation (SP), which is an efficient algorithm developed recently for analyzing the microscopic properties of glassy states for a fixed sample of disordered systems, can be reproduced by assuming the simplest replica symmetry on stationary points of the replicated Bethe free energy. Belief propagation and generalized SP can also be offered in the identical framework under assumptions of the highest and broken replica symmetries, respectively.