Markov degree of configurations defined by fibers of a configuration

Abstract

We consider a series of configurations defined by fibers of a given base configuration. We prove that Markov degree of the configurations is bounded from above by the Markov complexity of the base configuration. As important examples of base configurations we consider incidence matrices of graphs and study the maximum Markov degree of configurations defined by fibers of the incidence matrices. In particular we give a proof that the Markov degree for two-way transportation polytopes is three.

Exact Goodness‐of‐Fit Testing for the Ising Model

“…As shown by Diaconis & Sturmfels (1998), finding a Markov basis for a particular model is equivalent to finding generators for a specific ideal in a polynomial ring and can (in principle) be computed using Gröbner bases techniques. This spurred more research in algebraic statistics, see (Aoki et al, 2012;Drton et al, 2009;Hoṡten & Sullivant, 2007;Rapallo, 2003;Pistone & Rogantin, 2013) or more recently (Cai et al, 2014;Gross et al, 2016;Koyama et al, 2015;Rauh & Sullivant, 2016;Slavković et al, 2014). However, computing a Markov basis is often computationally intractable, making the Markov basis approach to goodness-of-fit testing ineffective in many applications.…”