Running Markov Chain Without Markov Bases

In this chapter we provide an algorithm for performing exact tests with a lattice basis, even in the case where Markov bases are not known. As mentioned in Sect. 15, computation of lattice bases is much easier than that of Markov bases. With many examples we show that the approach with lattice bases is practical. We also check that its performance is comparable to Markov bases for the problems where Markov bases are known. This chapter is based on Hara et al. (Proceedings of the Second CREST-SBM International Conference, “Harmony of Grobner Bases and the Modern Industrial Society”. World Scientfic, Singapore, 2012. To appear).

[1]  S. Sullivant,et al.  Sequential importance sampling for multiway tables , 2006, math/0605615.

[2]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[3]  Adrian Dobra,et al.  Dynamic Markov Bases , 2011, 1103.4891.

[4]  Satoshi Aoki,et al.  Markov Bases in Algebraic Statistics , 2012 .

[5]  Akimichi Takemura,et al.  On connectivity of fibers with positive marginals in multiple logistic regression , 2010, J. Multivar. Anal..

[6]  B. Sturmfels.  Gröbner bases and convex polytopes , 1995 .

[7]  Yuguo Chen,et al.  Sequential Monte Carlo Methods for Statistical Analysis of Tables , 2005 .

[8]  Guido Knapp,et al.  Conndence Intervals for the between Group Variance in the Unbalanced Oneeway Random Eeects Model of Analysis of Variance Journal of Statistical Computation and Simulation , 2007 .

[9]  P. Diaconis,et al.  Algebraic algorithms for sampling from conditional distributions , 1998 .

[10]  B. Sturmfels,et al.  Combinatorial Commutative Algebra , 2004 .

[11]  Dylan S. Small,et al.  Exact tests for the rasch model via sequential importance sampling , 2005 .

[12]  Adrian Dobra,et al.  LATTICE POINTS, CONTINGENCY TABLES, AND SAMPLING , 2007 .

[13]  Bernd Sturmfels,et al.  Higher Lawrence configurations , 2003, J. Comb. Theory, Ser. A.

[14]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[15]  Ruriko Yoshida,et al.  Algebraic and Geometric Methods in Statistics: Markov chains, quotient ideals and connectivity with positive margins , 2009 .