  Approximating marginals of a graphical model is one of the fundamental
problems in the theory of networks. In a recent paper a method was shown to
construct a variational free energy such that the linear response estimates,
and maximum entropy estimates (for beliefs) are in agreement, with implications
for direct and inverse Ising problems[1]. In this paper we demonstrate an
extension of that method, incorporating new information from the response
matrix, and we recover the adaptive-TAP equations as the first order
approximation[2]. The method is flexible with respect to applications of the
cluster variational method, special cases of this method include Naive Mean
Field (NMF) and Bethe. We demonstrate that the new framework improves
estimation of marginals by orders of magnitude over standard implementations in
the weak coupling limit. Beyond the weakly coupled regime we show there is an
improvement in a model where the NMF and Bethe approximations are known to be
poor for reasons of frustration and short loops.
