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We give a bijection between a quotient space of the parameters and the spaceof moments for any $A$-hypergeometric distribution. An algorithmic method tocompute the inverse image of the map is proposed utilizing the holonomicgradient method and an asymptotic equivalence of the map and the iterativeproportional scaling. The algorithm gives a method to solve a conditionalmaximum likelihood estimation problem in statistics. Our interplay between thetheory of hypergeometric functions and statistics gives some new formulas of$A$-hypergeometric polynomials.

$A$-Hypergeometric Distributions and Newton Polytopes