Exact tests for contingency tables and discrete exponential families -- Markov chain Monte Carlo methods over discrete sample space -- Toric ideals and their Gröbner bases -- Definition of Markov bases and other bases -- Structure of minimal Markov bases -- Method of distance reduction -- Symmetry of Markov bases -- Decomposable models of contingency tables -- Markov basis for no-three-factor interaction models and some other hierarchical models -- Two-way tables with structural zeros and fixed subtable sums -- Regular factorial designs with discrete response variables -- Group-wise selection models -- The set of moves connecting specific fibers -- Disclosure limitation problem and Markov basis -- Gröbner basis techniques for design of experiments -- Running Markov chain without Markov bases -- References -- Index.

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. Satoshi Aoki obtained his doctoral degree from University of Tokyo in 2004 and is currently an associate professor in Graduate school of Science and Engineering, Kagoshima University. Hisayuki Hara obtained his doctoral degree from University of Tokyo in 1999 and is currently an associate professor in Faculty of Economics, Niigata University. Akimichi Takemura obtained his doctoral degree from Stanford University in 1982 and is currently a professor in Graduate School of Information Science and Technology, University of Tokyo.