  We give a bijection between a quotient space of the parameters and the space
of moments for any $A$-hypergeometric distribution. An algorithmic method to
compute the inverse image of the map is proposed utilizing the holonomic
gradient method and an asymptotic equivalence of the map and the iterative
proportional scaling. The algorithm gives a method to solve a conditional
maximum likelihood estimation problem in statistics. Our interplay between the
theory of hypergeometric functions and statistics gives some new formulas of
$A$-hypergeometric polynomials.
