Overview
- Discusses the intersection of three subjects that are generally studied independently from each other: partitions, hypergeometric systems, and Dirichlet processes
- Explains the relationship between the above three subjects with simple problems that broaden readers’ mathematical horizons and statistical interests
- Provides an interdisciplinary approach that appeals to a wide audience, including statisticians, mathematicians, and researchers working in various fields of data sciences
Part of the book series: SpringerBriefs in Statistics (BRIEFSSTATIST)
Part of the book sub series: JSS Research Series in Statistics (JSSRES)
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About this book
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.
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Table of contents (5 chapters)
Authors and Affiliations
About the author
10-3, Midori-cho, Tachikawa, Tokyo 190-8562, Japan
Bibliographic Information
Book Title: Partitions, Hypergeometric Systems, and Dirichlet Processes in Statistics
Authors: Shuhei Mano
Series Title: SpringerBriefs in Statistics
DOI: https://doi.org/10.1007/978-4-431-55888-0
Publisher: Springer Tokyo
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: The Author(s) 2018
Softcover ISBN: 978-4-431-55886-6Published: 25 July 2018
eBook ISBN: 978-4-431-55888-0Published: 12 July 2018
Series ISSN: 2191-544X
Series E-ISSN: 2191-5458
Edition Number: 1
Number of Pages: VIII, 135
Number of Illustrations: 9 b/w illustrations
Topics: Statistical Theory and Methods, Statistics and Computing/Statistics Programs, Statistics for Engineering, Physics, Computer Science, Chemistry and Earth Sciences