Algebraic Statistics is, as the name suggests, the use of algebraic techniques in statistics. In recent years, this use has flourished, with algebra and algebraic geometry becoming useful in the study of things such as experimental design, parameter estimation, and hypothesis testing. At the same time, differential geometry has begun to show its uses in statistics, as exponential curvature has turned out to be a good measure for certain properties of estimation. The volume under review collects a number of papers in these areas and aims to be an introduction to the field while still containing advanced research topics. The table of contents gives a good ideal of the range of topics.
One can imagine two types of readers for this book: statisticians who want to learn how algebra can be used in their work and algebraists/geometers who want to learn about some new applications of their field. I cannot say how readers in the former camp would react to this book. I fall into the latter camp, and I must say that I did not find the book wholly successful. In particular, while several of the authors seemed to motivate and give background material on the algebra necessary for their results, fewer give background on the statistics, and even the articles that were designated as “xpository” tended to not give as much background as this reader needed. However, the authors did convince this reader that this is a fruitful area, and that more conversation between pure mathematicians and statisticians would be of great benefit to both.
Darren Glass is an Associate Professor of Mathematics at Gettysburg College, whose primary mathematical interests are algebraic geometry and number theory. He can be reached at firstname.lastname@example.org
List of contributors; Frequently used notations and symbols; Preface; 1. Algebraic and geometric methods in statistics P. Gibilisco, E. Riccomagno, M. P. Rogantin and H. P. Wynn; Part I. Contingency Tables: 2. Maximum likelihood estimation in latent class models S. E. Fienberg, P. Hersh, A. Rinaldo and Y. Zhou; 3. Algebraic geometry of 2 x 2 contingency tables A. Slavkovic and S. E. Fienberg; 4. Model selection for contingency tables with algebraic statistics A. Krampe and S. Kuhnt; 5. Markov chains, quotient ideals, and connectivity Y. Chen, I. Dinwoodie and R. Yoshida; 6. Algebraic category distinguishability E. Carlini and F. Rapallo; 7. Algebraic complexity of MLE for bivariate missing data S. Hoşten and S. Sullivant; 8. The generalized shuttle algorithm A. Dobra and S. E. Fienberg; Part II. Designed Experiments: 9. Generalised design H. Maruri-Aguilar and H. P. Wynn; 10. Design of experiments and biochemical network inference R. Laubenbacher and B. Stigler; 11. Replicated measurements and algebraic statistics R. Notari and E. Riccomagno; 12. Indicator function and sudoku designs R. Fontana and M. P. Rogantin; 13. Markov basis for design of experiments and three-level factors S. Aoki and A. Takemura; Part III. Information Geometry: 14. Non-parametric estimation R. F. Streater; 15. Banach manifold of quantum states R. F. Streater; 16. On quantum information manifolds A. Jenčová; 17. Axiomatic geometries for text documents G. Lebanon; 18. Exponential manifold by reproducing kernel Hilbert spaces K. Fukumizu; 19. Extended exponential models D. Imparato and B. Trivellato; 20. Quantum statistics and measures of quantum information F. Hansen; Part IV. Information Geometry and Algebraic Statistics: 21. Algebraic varieties vs differentiable manifolds G. Pistone; Part V. On-Line Supplements: Coloured Figures for Chapter 2; 22. Maximum likelihood estimation in latent class models Y. Zhou; 23. The generalized shuttle algorithm A. Dobra and S. E. Fienberg; 24. Indicator function and sudoku designs R. Fontana and M. P. Rogantin; 25. Replicated measurements and algebraic statistics R. Notari and E. Riccomagno; 26. Extended exponential models D. Imparato and B. Trivellato.