  We consider an interacting system of spin variables on a loopy interaction
graph, identified by a tree graph and a set of loopy interactions. We start
from a high-temperature expansion for loopy interactions represented by a sum
of nonnegative contributions from all the possible frustration-free loop
configurations. We then compute the loop corrections using different
approximations for the nonlocal loop interactions induced by the spin
correlations in the tree graph. For distant loopy interactions, we can exploit
the exponential decay of correlations in the tree interaction graph to compute
loop corrections within an independent-loop approximation. Higher orders of the
approximation are obtained by considering the correlations between the nearby
loopy interactions involving larger number of spin variables. In particular,
the sum over the loop configurations can be computed "exactly" by the belief
propagation algorithm in the low orders of the approximation as long as the
loopy interactions have a tree structure. These results might be useful in
developing more accurate and convergent message-passing algorithms exploiting
the structure of loopy interactions.
