Loopy belief propagation and Gibbs measures

We address the question of convergence in the loopy belief propagation (LBP) algorithm. Specifically, we relate convergence of LBP to the existence of a weak limit for a sequence of Gibbs measures defined on the LBP's associated computation tree. Using tools from the theory of Gibbs measures we develop easily testable sufficient conditions for convergence. The failure of convergence of LBP implies the existence of multiple phases for the associated Gibbs specification. These results give new insight into the mechanics of the algorithm.