Computing loop corrections by message passing

Any spanning tree in a loopy interaction graph can be used for communicating the effect of the loopy interactions by introducing messages that are passed along the edges in the spanning tree. This defines an exact mapping of the problem on the loopy interaction graph onto an extended problem on a tree interaction graph, where the thermodynamic quantities can be computed by a message-passing algorithm based on the Bethe equations. We propose an approximation loop correction algorithm for the Ising model relying on the above representation of the problem. The algorithm deals at the same time with the short and long loops, and can be used to obtain upper and lower bounds for the free energy.