This book focuses on statistical inferences related to various combinatorial stochastic
processes. Specifically it discusses the intersection of three subjects that are generally
studied independently of each other: partitions hypergeometric systems and Dirichlet
processes. The Gibbs partition is a family of measures on integer partition and several prior
processes such as the Dirichlet process naturally appear in connection with infinite
exchangeable Gibbs partitions. Examples include the distribution on a contingency table with
fixed marginal sums and the conditional distribution of Gibbs partition given the length. The
A-hypergeometric distribution is a class of discrete exponential families and appears as the
conditional distribution of a multinomial sample from log-affine models. The normalizing
constant is the A-hypergeometric polynomial which is a solution of a system of linear
differential equations of multiple variables determined by a matrix A called A-hypergeometric
system. The book presents inference methods based on the algebraic nature of the
A-hypergeometric system and introduces the holonomic gradient methods which numerically solve
holonomic systems without combinatorial enumeration to compute the normalizing constant.
Furher it discusses Markov chain Monte Carlo and direct samplers from A-hypergeometric
distribution as well as the maximum likelihood estimation of the A-hypergeometric distribution
of two-row matrix using properties of polytopes and information geometry. The topics discussed
are simple problems but the interdisciplinary approach of this book appeals to a wide audience
with an interest in statistical inference on combinatorial stochastic processes including
statisticians who are developing statistical theories and methodologies mathematicians wanting
to discover applications of their theoretical results and researchers working in various
fields of data sciences.
