Algebraic statistics is a rapidly developing field, where ideas from statistics and algebra meet and stimulate new research directions. One of the origins of algebraic statistics is the work by Diaconis and Sturmfels in 1998 on the use of Gröbner bases for constructing a connected Markov chain for performing conditional tests of a discrete exponential family. In this book we take up this topic and present a detailed summary of developments following the seminal work of Diaconis and Sturmfels.
This book is intended for statisticians with minimal backgrounds in algebra. As we ourselves learned algebraic notions through working on statistical problems and collaborating with notable algebraists, we hope that this book with many practical statistical problems is useful for statisticians to start working on the field.
Author(s): Satoshi Aoki, Hisayuki Hara, Akimichi Takemura (auth.)
Series: Springer Series in Statistics 199
Edition: 1
Publisher: Springer-Verlag New York
Year: 2012
Language: English
Pages: 300
City: New York
Tags: Statistics, general; Statistical Theory and Methods; General Algebraic Systems; Applications of Mathematics
Front Matter....Pages i-xi
Front Matter....Pages 1-1
Exact Tests for Contingency Tables and Discrete Exponential Families....Pages 3-21
Markov Chain Monte Carlo Methods over Discrete Sample Space....Pages 23-31
Toric Ideals and Their Gröbner Bases....Pages 33-43
Front Matter....Pages 45-45
Definition of Markov Bases and Other Bases....Pages 47-63
Structure of Minimal Markov Bases....Pages 65-78
Method of Distance Reduction....Pages 79-89
Symmetry of Markov Bases....Pages 91-105
Front Matter....Pages 107-107
Decomposable Models of Contingency Tables....Pages 109-128
Markov Basis for No-Three-Factor Interaction Models and Some Other Hierarchical Models....Pages 129-157
Two-Way Tables with Structural Zeros and Fixed Subtable Sums....Pages 159-179
Regular Factorial Designs with Discrete Response Variables....Pages 181-208
Groupwise Selection Models....Pages 209-227
The Set of Moves Connecting Specific Fibers....Pages 229-247
Front Matter....Pages 249-249
Disclosure Limitation Problem and Markov Basis....Pages 251-259
Gröbner Basis Techniques for Design of Experiments....Pages 261-273
Running Markov Chain Without Markov Bases....Pages 275-286
Back Matter....Pages 287-298
