  Near-Gaussian probability densities are common in many important physical
applications. Here we develop an asymptotic expansion methodology for computing
entropic functionals for such densities. The expansion proposed is a close
relative of standard perturbation expansions in quantum field theory. We give
novel results on the low-order effects of non-Gaussian even moments and
asymmetry (e.g. skewness) on the entropy. The asymptotic expansion is also used
to define a best fit maximum entropy density given a set of observed low order
moments. The maximum entropy density estimation technique consists simply of
the solution of a small set of algebraic equations and is therefore more
straightforward numerically than classical maximum-entropy methods which rely
on sophisticated convex optimization techniques.
