Markov degrees of hierarchical models and Betti numbers of Stanley-Reisner ideals

There are two seemingly unrelated classical objects associated to a simplicial complex: a hierarchical model and a Stanley-Reisner ring. A hierarchical model gives rise to a toric ideal, a relationship that is a staple of algebraic statistics. In this note, we explore the connection between degrees of Markov bases elements of the model and the rows of the Betti diagram of the Stanley-Reisner ideal. We propose a precise conjecture, which we establish in several cases, most notably for decomposable and vertex-decomposable complexes. In turn, this connection implies the following for complexes satisfying our conjecture: if the ideal of the hierarchical model is generated in one degree, then the Stanley-Reisner ring of the complex has a linear resolution over any field. In particular, this holds for clique complexes of chordal graphs: those corresponding to decomposable graphical models, whose Markov bases are known to be quadratic.