Open Problems on Connectivity of Fibers with Positive Margins in Multi-dimensional Contingency Tables

Abstract

Abstract. Diaconis-Sturmfels developed an algorithm for sampling from conditional distributionsfor a statistical model of discrete exponential families, based on the algebraic theory of toricideals. This algorithm is applied to categorical data analysis through the notion of Markov bases.Initiated with its application to Markov chain Monte Carlo approach for testing statistical fittingof the given model, many researchers have extensively studied the structure of Markov bases formodels in computational algebraic statistics. In the Markov chain Monte Carlo approach fortesting statistical fitting of the given model, a Markov basis is a set of moves connecting allcontingency tables satisfying the given margins. Despite the computational advances, there areapplied problems where one may never be able to compute a Markov basis. In general, the numberof elements in a minimal Markov basis for a model can be exponentially many. Thus, it is importantto compute a reduced number of moves which connect all tables instead of computing a Markovbasis. In some cases, such as logistic regression, positive margins are shown to allow a set ofMarkov connecting moves that are much simpler than the full Markov basis. Such a set is calleda Markov subbasis with assumption of positive margins.In this paper we summarize some computations of and open problems on Markov subbases forcontingency tables with assumption of positive margins under specific models as well as developalgebraic methods for studying connectivity of Markov moves with margin positivity to developMarkov sampling methods for exact conditional inference in statistical models where the Markovbasis is hard to compute.