The A-hypergeometric system of associated with the rational normal curve and statistical inference of exchangeable combinatorial structures

The A-hypergeometric polynomial associated with the rational normal curve is the constant multiple of the associated partial Bell polynomials. The distribution whose normalization constant is the partial Bell polynomials is an algebraic exponential family, known as the microcanonical Gibbs distribution. It is a model of exchangeable random combinatorial structures. Algebraic methods to evaluate the associated partial Bell polynomials numerically with the Pfaffian system, which are called the holonomic gradient methods, are presented. Especially, the maximum likelihood estimation (MLE) of the full and the curved exponential families are studied in terms of the information geometry of the Newton polytope. It is shown that the MLE does not exist for the full exponential family.