Algebraic and Geometric Methods in Statistics

doi:10.1017/CBO9780511642401

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.

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Contents

Frontmatter
pp i-vi
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Contents
pp vii-viii
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List of contributors
pp ix-xii
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Preface
pp xiii-xv
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Frequently used notations and symbols
pp xvi-xvi
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1 - Algebraic and geometric methods in statistics
pp 1-24
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Part I - Contingency tables
pp 25-26
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2 - Maximum likelihood estimation in latent class models for contingency table data
pp 27-62
- By S. E. Fienberg, P. Hersh, A. Rinaldo, Y. Zhou
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3 - Algebraic geometry of 2×2 contingency tables
pp 63-82
- By A. B. Slavković, S. E. Fienberg
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4 - Model selection for contingency tables with algebraic statistics
pp 83-98
- By A. Krampe, S. Kuhnt
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5 - Markov chains, quotient ideals and connectivity with positive margins
pp 99-110
- By Y. Chen, I. Dinwoodie, R. Yoshida
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6 - Algebraic modelling of category distinguishability
pp 111-122
- By E. Carlini, F. Rapallo
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7 - The algebraic complexity of maximum likelihood estimation for bivariate missing data
pp 123-134
- By S. Hossten, S. Sullivant
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8 - The generalised shuttle algorithm
pp 135-156
- By A. Dobra, S. E. Fienberg
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Part II - Designed experiments
pp 157-158
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9 - Generalised design: interpolation and statistical modelling over varieties
pp 159-174
- By H. Maruri-Aguilar, H. P. Wynn
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10 - Design of experiments and biochemical network inference
pp 175-186
- By R. Laubenbacher, B. Stigler
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11 - Replicated measurements and algebraic statistics
pp 187-202
- By R. Notari, E. Riccomagno
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12 - Indicator function and sudoku designs
pp 203-224
- By R. Fontana, M. P. Rogantin
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13 - Markov basis for design of experiments with three-level factors
pp 225-238
- By S. Aoki, A. Takemura
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Part III - Information geometry
pp 239-240
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14 - Introduction to non-parametric estimation
pp 241-256
- By R. F. Streater
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15 - The Banach manifold of quantum states
pp 257-264
- By R. F. Streater
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16 - On quantum information manifolds
pp 265-276
- By A. Jenčová
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17 - Axiomatic geometries for text documents
pp 277-290
- By G. Lebanon
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18 - Exponential manifold by reproducing kernel Hilbert spaces
pp 291-306
- By K. Fukumizu
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19 - Geometry of extended exponential models
pp 307-326
- By D. Imparato, B. Trivellato
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20 - Quantum statistics and measures of quantum information
pp 327-338
- By F. Hansen
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Part IV - Information geometry and algebraic statistics
pp 339-340
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21 - Algebraic varieties vs. differentiable manifolds in statistical models
pp 341-366
- By G. Pistone
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Algebraic and Geometric Methods in Statistics

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