First International School on Algebraic Statistics STID – Menton ( France ) February , 17 - 18 , 2003 Toric Statistical Models

In this tutorial we show the practical applicability of the algorithm described in the paper by Diaconis & Sturmfels (1998) who apply the algebraic theory of toric ideals to define a Markov Chain Monte Carlo method for sampling from conditional distributions, through the notion of Markov basis. For some basic concepts from Computational Commutative Algebra, such as polynomial ideal, term-ordering and Gröbner basis, we refer to Kreuzer & Robbiano (2000). Fundamental references for the applications of Commutative Algebra to Probability and Statistics for finite sample spaces are Pistone et al. (2001a), Chapter 6, and Pistone et al. (2001b). For the theory of log-linear models, we can refer to Fienberg (1980) and Agresti (2002). Let X be a finite set and let px = P[x] = φ(T (x)) x ∈ X , (1)