The Zariski closure of a discrete log-linear statistical model is a toric variety. In the last twenty years, this fact has lead not only to the development of useful algorithms for working with data, but developments in algebraic geometry and combinatorics as well. I will discuss a particular subset of the discrete log-linear models known as hierarchical models. Such models are naturally indexed by simplicial complexes whose vertices are labeled by positive integers. I will present some results on how one can infer certain properties of such a model based on the combinatorics of its associated vertex-labeled simplicial complex. Results given come from joint work with Christopher O’Neill and Seth Sullivant. This is a practice run of my thesis defense.