The cavity method has been developed to investigate properties of dilute spin glasses, or, which is largely the same thing, properties of sets of solutions of (large) random constraint satisfaction problems. The cavity method has, as usually presented, similarities to Belief Propagation, but a systematic derivation has been lacking.
In a lecture at a Statphys satellite meeting this summer, Marc Mezard presented the cavity method as a special solution to a Belief Propagation on a probability space consisting of the solutions to ordinary Belief Propagation, weighted by the Boltzmann factor derived from the Yedidia-Weiss-Freeman free energy (that is, the Bethe ansatz in this context).
I will describe this theory, and present inconclusive results whether not the cavity method is the the only reasonable solution to Mezard's second order Belief Propagation.
