  The success of the "Cluster Variation Method" (CVM) in reproducing quite
accurately the free energies of Monte Carlo (MC) calculations on Ising models
is explained in terms of identifying a cancellation of errors: We show that the
CVM produces correlation functions that are too close to zero, which leads to
an overestimation of the exact energy, E, and at the same time, to an
underestimation of -TS, so the free energy F=E-TS is more accurate than either
of its parts. This insight explains a problem with "hybrid methods" using MC
correlation functions in the CVM entropy expression: They give exact energies E
and do not give significantly improved -TS relative to CVM, so they do not
benefit from the above noted cancellation of errors. Additionally, "hybrid
methods" suffer from the difficulty of adequately accounting for both ordered
and disordered phases in a consistent way. A different technique, the "Entropic
Monte Carlo" (EMC), is shown here to provide a means for critically evaluating
the CVM entropy. Inspired by EMC results, we find a universal and simple
correction to the CVM entropy which produces individual components of the free
energy with MC accuracy, but is computationally much less expensive than either
MC thermodynamic integration or EMC.
