Partitions, hypergeometric systems, and Dirichlet processes in statistics
type: Item
instance of: scholarly article
title: Partitions, hypergeometric systems, and Dirichlet processes in statistics
zbMATH Open document ID: 1404.62003
DOI: 10.1007/978-4-431-55888-0
author: Shuhei Mano
published in: SpringerBriefs in Statistics
publication date: 2018-08-20T00:00:00Z
review text: This short monograph presents key ideas in the study of random partitions, hypergeometric systems and Dirichlet processes, and in particular the links between them, in the context of statistical inference, taking advantage of the underlying exchangeability present in these models. The first chapter gives a short (and very readable) introduction to the key ideas that will be needed through the rest of the book: partitions and exchangeability. Two simple motivating examples are then given, to help the reader understand the goal of the work before starting the more technical material presented in Chapters 2--5. The first example is a two-cluster Bayesian mixture model, and the second a problem of testing goodness-of-fit in a Poisson regression model. Chapter 2 discusses probability measures on combinatorial structures. The author begins by introducing measures on partitions, and in particular the multiplicative measures and combinatorial structures associated with binomial, negative binomial and Poisson coefficients: powersets, multisets and assemblies. Some examples are given (for example, integer partitions with or without the constraint that the elements in the partition be distinct) with clear references to lead the reader to other sources of examples. Exponential structures are then discussed, and examples are again provided (such as set partitions, where the numbers of possible instances are given by the Bell numbers). There is some brief discussion of statistical applications, including to the first of the motivating examples of Chapter 1. Tilting of probability measures and partial Bell polynomials are introduced (again, accompanied by examples), before the chapter concludes with asymptotic results on the size of extreme (both large and small) parts of Gibbs partitions, in particular for the Pitman partition and Ewens sampling formula. The main object of Chapter 3 is to introduce the \(A\)-hypergeometric system and distribution for a nonnegative, integer-valued…
reviewed by: Fraser Daly
Mathematics Subject Classification ID: 62-02
Mathematics Subject Classification ID: 60C05
Mathematics Subject Classification ID: 60G09
Mathematics Subject Classification ID: 60G57
Mathematics Subject Classification ID: 62F15
Mathematics Subject Classification ID: 62G05
Mathematics Subject Classification ID: 33D90
Mathematics Subject Classification ID: 62H30
Mathematics Subject Classification ID: 60G44
zbMATH DE Number: 6921105
zbMATH Keywords: partition
zbMATH Keywords: combinatorial structure
zbMATH Keywords: exchangeability
zbMATH Keywords: hypergeometric systems
zbMATH Keywords: Dirichlet process
zbMATH Keywords: algebraic statistics
zbMATH Keywords: Bayesian statistics
zbMATH Keywords: Ewens sampling formula
zbMATH Keywords: mixture model
zbMATH Keywords: martingale limit theorem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-4-431-55888-0
OpenAlex ID: W2870847272
description: scientific article; zbMATH DE number 6921105