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Partitions, Hypergeometric Systems, and Dirichlet Processes in Statistics

  • Shuhei¬†Mano
Book

Part of the SpringerBriefs in Statistics book series (BRIEFSSTATIST)

Also part of the JSS Research Series in Statistics book sub series (JSSRES)

Table of contents

  1. Front Matter
    Pages i-viii
  2. Shuhei Mano
    Pages 1-9
  3. Shuhei Mano
    Pages 11-43
  4. Shuhei Mano
    Pages 45-70
  5. Shuhei Mano
    Pages 71-103
  6. Shuhei Mano
    Pages 105-122
  7. Back Matter
    Pages 123-135

About this book

Introduction

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.

Keywords

Combinatorial Stochastic Processes Random Combinatorial Models Random Combinatorial Structures Statistical Inferences Stochastic Models

Authors and affiliations

  • Shuhei¬†Mano
    • 1
  1. 1.The Institute of Statistical MathematicsTachikawaJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-4-431-55888-0
  • Copyright Information The Author(s) 2018
  • Publisher Name Springer, Tokyo
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-4-431-55886-6
  • Online ISBN 978-4-431-55888-0
  • Series Print ISSN 2191-544X
  • Series Online ISSN 2191-5458
  • Buy this book on publisher's site