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Zbl 0952.62088
Diaconis, Persi; Sturmfels, Bernd
Algebraic algorithms for sampling from conditional distributions. (English)
[J] Ann. Stat. 26, No.1, 363-397 (1998). ISSN 0090-5364

This paper aims to describe the construction of new Markov chain algorithms for sampling from discrete exponential families conditional on a sufficient statistic. Section 2 introduces the necessary stochastic and statistical background. Section 3 contains the main contribution of the paper: it shows how to compute a Markov basis using tools from computational algebra. More precisely, to find a Markov basis is proved to be equivalent to finding a set of generators of an ideal in a polynomial ring, using Gröbner bases. To represent this ideal in a way suitable for computation is illustrated by MATHEMATICA and MAPLE programs. The next sections of the paper include detailed treatments of the proposed technique for some important special cases: contingency tables (Section 4), logistic regression (in Section 5), and the spectral analysis of permutation data (Section 6).
[Neculai Curteanu (Iaşi)]

MSC 2000:: *62M99 Inference from stochastic processes
65C60 Computational problems in statistics
13P10 Polynomial ideals, Groebner bases
62F25 Parametric confidence regions, etc.
62H17 Contingency (statistics)

Keywords: conditional distribution; computational algebra methods; Markov chain algorithms for sampling; Markov basis; contingency tables; logistic regression; spectral analysis of permutation data

Cited in: Zbl 1206.62117 Zbl 1089.62136 Zbl 1076.62060 Zbl 1049.62068

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