  We present a new implementation of the Cluster Variational Method (CVM) as a
message passing algorithm. The kind of message passing algorithms used for CVM,
usually named Generalized Belief Propagation, are a generalization of the
Belief Propagation algorithm in the same way that CVM is a generalization of
the Bethe approximation for estimating the partition function. However, the
connection between fixed points of GBP and the extremal points of the CVM
free-energy is usually not a one-to-one correspondence, because of the
existence of a gauge transformation involving the GBP messages.
  Our contribution is twofold. Firstly we propose a new way of defining
messages (fields) in a generic CVM approximation, such that messages arrive on
a given region from all its ancestors, and not only from its direct parents, as
in the standard Parent-to-Child GBP. We call this approach maximal messages.
Secondly we focus on the case of binary variables, re-interpreting the messages
as fields enforcing the consistency between the moments of the local (marginal)
probability distributions. We provide a precise rule to enforce all
consistencies, avoiding any redundancy, that would otherwise lead to a gauge
transformation on the messages. This moment matching method is gauge free, i.e.
it guarantees that the resulting GBP is not gauge invariant.
  We apply our maximal messages and moment matching GBP to obtain an analytical
expression for the critical temperature of the Ising model in general
dimensions at the level of plaquette-CVM. The values obtained outperform Bethe
estimates, and are comparable with loop corrected Belief Propagation equations.
The method allows for a straightforward generalization to disordered systems.
