  We develop the method of Maximum Entropy (ME) as a technique to generate
approximations to probability distributions. The central results consist in (a)
justifying the use of relative entropy as the uniquely natural criterion to
select a "best" approximation from within a family of trial distributions, and
(b) to quantify the extent to which non-optimal trial distributions are ruled
out. The Bogoliuvob variational method is shown to be included as a special
case. As an illustration we apply our method to simple fluids. In a first use
of the ME method the "exact" canonical distribution is approximated by that of
a fluid of hard spheres and ME is used to select the optimal value of the
hard-sphere diameter. A second, more refined application of the ME method
approximates the "exact" distribution by a suitably weighed average over
different hard-sphere diameters and leads to a considerable improvement in
accounting for the soft-core nature of the interatomic potential. As a specific
example, the radial distribution function and the equation of state for a
Lennard-Jones fluid (Argon) are compared with results from molecular dynamics
simulations.
