  A general mean field theory is presented for the construction of equilibrium
coarse grained models. Inverse methods that reconstruct microscopic models from
low resolution experimental data can be derived as particular implementations
of this theory. The theory also applies to the opposite problem of reduction,
where relevant information is extracted from available equilibrium ensemble
data. These problems are central to the construction of coarse grained
representations of complex systems, and commonly used coarse graining methods
are derived as particular cases of the general theory.
