  Approximations of loopy belief propagation, including expectation propagation
and approximate message passing, have attracted considerable attention for
probabilistic inference problems. This paper proposes and analyzes a
generalization of Opper and Winther's expectation consistent (EC) approximate
inference method. The proposed method, called Generalized Expectation
Consistency (GEC), can be applied to both maximum a posteriori (MAP) and
minimum mean squared error (MMSE) estimation. Here we characterize its fixed
points, convergence, and performance relative to the replica prediction of
optimality.
