  A loop series expansion for the partition function of a general statistical
model on a graph is carried out. If the auxiliary probability distributions of
the expansion are chosen to be a fixed point of the belief-propagation
equation, the first term of the loop series gives the Bethe-Peierls free energy
functional at the replica-symmetric level of the mean-field spin glass theory,
and corrections are contributed only by subgraphs that are free of dangling
edges. This result generalize the early work of Chertkov and Chernyak on binary
statistical models. If the belief-propagation equation has multiple fixed
points, a loop series expansion is performed for the grand partition function.
The first term of this series gives the Bethe-Peierls free energy functional at
the first-step replica-symmetry-breaking (RSB) level of the mean-field
spin-glass theory, and corrections again come only from subgraphs that are free
of dangling edges, provided that the auxiliary probability distributions of the
expansion are chosen to be a fixed point of the survey-propagation equation.
The same loop series expansion can be performed for higher-level partition
functions, obtaining the higher-level RSB Bethe-Peierls free energy functionals
(and the correction terms) and message-passing equations without using the
Bethe-Peierls approximation.
