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.