Running Markov chain without Markov basis

TL;DR

This paper introduces a practical method for performing exact tests using lattice bases instead of Markov bases, which are difficult to compute for complex models, demonstrating comparable performance and broader applicability.

Contribution

The paper proposes using lattice bases as an alternative to Markov bases for exact tests, enabling analysis of models where Markov bases are unknown or hard to compute.

Findings

Lattice bases are easier to compute than Markov bases.

The approach is practical for complex models.

Performance is comparable to Markov bases where known.

Abstract

The methodology of Markov basis initiated by Diaconis and Sturmfels(1998) stimulated active research on Markov bases for more than ten years. It also motivated improvements of algorithms for Grobner basis computation for toric ideals, such as those implemented in 4ti2. However at present explicit forms of Markov bases are known only for some relatively simple models, such as the decomposable models of contingency tables. Furthermore general algorithms for Markov bases computation often fail to produce Markov bases even for moderate-sized models in a practical amount of time. Hence so far we could not perform exact tests based on Markov basis methodology for many important practical problems. In this article we propose to use lattice bases for performing exact tests, in the case where Markov bases are not known. 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.