Belief propagation (BP) on cyclic graphs is an efficient algorithm for computing approximate marginal probability distributions over single nodes and neighboring nodes in the graph. It does however not prescribe a way to compute joint distributions over pairs of distant nodes in the graph. In this paper we propose two new algorithms for approximating these pairwise probabilities, based on the linear response theorem. The first is a propagation algorithm which is shown to converge if belief

more »

... gation converges to a stable fixed point. The second algorithm is based on matrix inversion. Applying these ideas to Gaussian random fields we derive a propagation algorithm for computing the inverse of a matrix.