Algebraic and Geometric Methods in Statistics

by Edited by Paolo Gibilisco , Eva Riccomagno , Maria Piera Rogantin , Henry P. Wynn

• ISBN13: 9780521896191
• ISBN10: 0521896193
• Edition: 1st
• Format: Hardcover
• Copyright: 2009-11-30
• Publisher: Cambridge University Press

Summary

This up-to-date account of algebraic statistics and information geometry explores the emerging connections between the two disciplines, demonstrating how they can be used in design of experiments and how they benefit our understanding of statistical models, in particular, exponential models. This book presents a new way of approaching classical statistical problems and raises scientific questions that would never have been considered without the interaction of these two disciplines. Beginning with a brief introduction to each area, using simple illustrative examples, the book then proceeds with a collection of reviews and some new results written by leading researchers in their respective fields. Part III dwells in both classical and quantum information geometry, containing surveys of key results and new material. Finally, Part IV provides examples of the interplay between algebraic statistics and information geometry. Computer code and proofs are also available online, where key examples are developed in further detail.

Table of Contents

List of contributors	p. ix
Preface	p. xiii
Frequently used notations and symbols	p. xvi
Algebraic and geometric methods in statistics The editors	p. 1
Contingency tables	p. 25
Maximum likelihood estimation in latent class models for contingency table data	p. 27
Algebraic geometry of 2x2 contingency tables	p. 63
Model selection for contingency tables with algebraic statistics	p. 83
Markov chains, quotient ideals and connectivity with positive margins	p. 99
Algebraic modelling of category distinguishability	p. 111
The algebraic complexity of maximum likelihood estimation for bivariate missing data	p. 123
The generalised shuttle algorithm	p. 135
Designed experiments	p. 157
Generalised design: interpolation and statistical modelling over varieties	p. 159
Design of experiments and biochemical network inference	p. 175
Replicated measurements and algebraic statistics	p. 187
Indicator function and sudoku designs	p. 203
Markov basis for design of experiments with three-level factors	p. 225
Information geometry	p. 239
Introduction to non-parametric estimation	p. 241
The Banach manifold of quantum states	p. 257
On quantum information manifolds	p. 265
Axiomatic geometries for text documents	p. 277
Exponential manifold by reproducing kernel Hilbert spaces	p. 291
Geometry of extended exponential models	p. 307
Quantum statistics and measures of quantum information	p. 327
Information geometry and algebraic statistics	p. 339
Algebraic varieties vs. differentiable manifolds in statistical models	p. 341
On-line supplements (available for download from www.cambridge.org/9780521S96191)	p. 367
Coloured figures for Chapter 2	p. 369
Maximum likelihood estimation in latent class models for contingency table data	p. 373
The generalised shuttle algorithm	p. 395
Indicator function and sudoku designs	p. 408
Replicated measurements and algebraic statistics	p. 424
Geometry of extended exponential models	p. 427
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