  We investigate families of generalized mean--field theories that can be
formulated using the Peierls--Bogoliubov inequality. For test--Hamiltonians
describing mutually non--interacting subsystems of increasing size, the
thermodynamics of these mean--field type systems approaches that of the
infinite, fully interacting system except in the immediate vicinity of their
respective mean--field critical points. Finite--size scaling analysis of this
mean--field critical behaviour allows to extract the critical exponents of the
fully interacting system. It turns out that this procedure amounts to the
coherent anomaly method (CAM) proposed by Suzuki, which is thus given a
transparent interpretation in terms of conventional renormalization group
ideas. Moreover, given the geometry of approximating systems, we can identify
the family of approximants which is optimal in the sense of the
Peierls--Bogoliubov inequality. In the case of the 2--$d$ Ising model it turns
out that, surprisingly, this optimal family gives rise to a spurious
singularity of thermodynamic functions.
