  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.
