Diaconis-Sturmfels showed that a set of binomial generators of a toric ideal for a statistical model of discrete exponential families is equivalent to a Markov basis and initiated Markov chain Monte Carlo approach based on a Groebner basis computation for testing statistical fitting of the given model, many researchers have extensively studied the structure of Markov bases for models in computational algebraic statistics. Despite the computational advances, there are applied problems where one may never be able to compute a Markov basis. In general, the number of elements in a minimal Markov basis for a model can be exponentially many. Thus, it is important to compute a reduced number of moves which connect all tables instead of computing a Markov basis. In some cases, such as logistic regression, positive margins are shown to allow a set of Markov connecting moves that are much simpler than the full Markov basis.
In this talk, we will present algebraic methods for studying connectivity of Markov moves with margin positivity. The purpose is to develop Markov sampling methods for exact conditional inference in statistical models where a Markov basis is hard to compute. In some cases positive margins are shown to allow a set of Markov connecting moves that are much simpler than the full Markov basis.