Summary: Markov bases first appeared in [P. Diaconis and B. Sturmfels, Ann. Stat. 26, No. 1, 363–397 (1998; Zbl 0952.62088)]. In this paper, they considered the problem of estimating the \(p\) values for conditional tests for data summarized in contingency tables by Markov chain Monte Carlo methods; this is one of the fundamental problems in applied statistics. In this setting, it is necessary to have an appropriate connected Markov chain over the given finite sample space. Diaconis and Sturmfels formulated this problem with the idea of a Markov basis, and they showed that it corresponds to the set of generators of a well-specified toric ideal. Their work is very attractive because the theory of a Gröbner basis, a concept of pure mathematics, can be used in actual problems in applied statistics. In fact, their work became one of the origins of the relatively new field, computational algebraic statistics. In this chapter, we first introduce their work along with the necessary background in statistics. After that, we use the theory of Gröbner bases to solve actual applied statistical problems in experimental design.
For the entire collection see [Zbl 1282.13003].