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Lectures on algebraic statistics. (English) Zbl 1166.13001
Oberwolfach Seminars 39. Basel: Birkhäuser (ISBN 978-3-7643-8904-8/pbk). viii, 271 p. (2009).
The seminar lectures give an overview of the present status of the research area in which techniques originating in algebraic geometry, commutative algebra and combinatorics are adapted to solve problems of interest in statistics. The contents of the book is divided in seven chapters. The first five chapters are expanded versions of lectures given at an Oberwolfach Seminar in May 2008: Markov Bases, Likelihood Inference, Conditional Independence, Hidden Variables, Bayesian Integrals. Each of them is devoted to an important model/problem of statistics and explains ties between them and notions/results of algebraic geometry/computational algebra. The sixth chapter contains students’ solutions to eight problems proposed during the Oberwolfach Seminar. In the final chapter one finds twelve open research problems related to algebraic statistics, presented by lecturers and students.
The authors have striven to gather as many facts as possible and to organize the large amount of material in a most profitable way for the astute reader. The general idea is that statistical hypotheses can be often tested in an exact approach by using algebraic tools developed to solve (semi-)algebraic equations. Many examples discussed at length in these lecture notes indicates that, in order to draw conclusions with practical applicability from a geometric study of the parameter space, heavy computations are unavoidable. Several software packages are used to perform such demanding computations. The necessary pieces of code are carefully explained.
The volume makes a difficult reading by the wealth of techniques and results touched upon. It is not self-contained. The reader is warned that to study all the material, at least as much enthusiasm and energy as that witnessed by the participants to the seminar are needed.

MSC:
13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
62-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to statistics
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
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