Abstract
In this paper we given some basic characterizations of minimal Markov basis for a connected Markov chain, which is used for performing exact tests in discrete exponential families given a sufficient statistic. We also give a necessary and sufficient condition for uniqueness of minimal Markov basis. A general algebraic algorithm for constructing a connected Markov chain was given by Diaconis and Sturmfels (1998,The Annals of Statistics,26, 363–397). Their algorithm is based on computing Gröbner basis for a certain ideal in a polynomial ring, which can be carried out by using available computer algebra packages. However structure and interpretation of Gröbner basis produced by the packages are sometimes not clear, due to the lack of symmetry and minimality in Gröbner basis computation. Our approach clarifies partially ordered structure of minimal Markov basis.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aoki, S. (2002). Improving path trimming in a network algorithm for Fisher’s exact test in two-way contingency tables,Journal of Statistical Computation and Simulation,72, 205–216.
Aoki, S. (2003). Network algorithm for the exact test of Hardy-Weinberg proportion for multiple alleles,Biometrical Journal,45, 471–490.
Aoki, S. and Takemura, A. (2002). Markov chain Monte Carlo exact tests for incomplete two-way contingency tables, Tech. Report, METR 02-08, Department of Mathematical Engineering and Information Physics, The University of Tokyo (submitted for publication).
Aoki, S. and Takemura, A. (2003). Minimal basis for connected Markov chain over 3×3×K contingency tables with fixed two-dimensional marginals,Australian and New Zealand Journal of Statistics,45, 229–249.
Bishop, Y. M. M., Fienberg, S. E. and Holland, P. W. (1975).Discrete Multivariate Analysis: Theory and Practice, The MIT Press, Cambridge, Massachusetts.
Briales, E., Campillo, A., Marijuán, C. and Pisón, P. W. (1975). Minimal systems of generators for ideals of semigroups,Journal of Pure and Applied Algebra,124, 7–30.
Diaconis, P. and Sturmfels, B. (1998). Algebraic algorithms for sampling from conditional distributions,The Annals of Statistics,26, 363–397.
Diaconis, P., Eisenbud, D. and Sturmfels, B. (1998). Lattice walks and primary decomposition,Mathematical Essays in Honor of Gian-Carlo Rota (eds. B. E. Sagan and R. P. Stanley), 173–193, Birkhauser, Boston.
Dinwoodie, J. H. (1998). The Diaconis-Sturmfels algorithm and rules of succession,Bernoulli,4, 401–410.
Dobra, A. and Sullivant, S. (2002). A divide-and-conquer algorithm for generating Markov bases of multi-way tables, National Institute of Statistical Sciences Tech. Report, No. 124 (available from http://www.niss.org/downloadabletechreports.html).
Guo, S. W. and Thompson, E. A. (1992). Performing the exact test of Hardy-Weinberg proportion for multiple alleles,Biometrika,48, 361–372.
Lauritzen, S. L. (1996).Graphical Models, Oxford University Press, Oxford.
Mehta, C. R. and Patel, N. R. (1983). A network algorithm for performing Fisher’s exact test inr×c contingency tables,Journal of The American Statistical Association,78, 427–434.
Sakata, T. and Sawae, R. (2000). Echelon form and conditional test for three way contingency tables,Proceedings of the 10-th Japan and Korea Joint Conference of Statistics, 333–338.
Schrijver, A. (1986).Theory of Linear and Integer Programming, Wiley, Chichester.
Wilson, R. J. (1985).Introduction to Graph Theory, 3rd ed., Longman, New York.
Author information
Authors and Affiliations
About this article
Cite this article
Takemura, A., Aoki, S. Some characterizations of minimal Markov basis for sampling from discrete conditional distributions. Ann Inst Stat Math 56, 1–17 (2004). https://doi.org/10.1007/BF02530522
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02530522