Toric Ideals of Characteristic Imsets via Quasi-Independence Gluing

Characteristic imsets are 0-1 vectors which correspond to Markov equivalence classes of directed acyclic graphs. The study of their convex hull, named the characteristic imset polytope, has led to new and interesting geometric perspectives on the important problem of causal discovery. In this paper we begin the study of the associated toric ideal. We develop a new generalization of the toric fiber product, which we call a quasi-independence gluing, and show that under certain combinatorial homogeneity conditions, one can iteratively compute a Gr\"obner basis via lifting. For faces of the characteristic imset polytope associated to trees, we apply this technique to compute a Gr\"obner basis for the associated toric ideal. We end with a study of the characteristic ideal of the cycle and propose directions for future work.Comment: 19 pages, 7 figure

Markov random fields and iterated toric fibre products

We prove that iterated toric fibre products from a finite collection of toric varieties are defined by binomials of uniformly bounded degree. This implies that Markov random fields built up from a finite collection of finite graphs have uniformly bounded Markov degree.Comment: several improvements, final versio

Tropical Geometry of T-Varieties with Applications to Algebraic Statistics

Varieties with group action have been of interest to algebraic geometers for centuries. In particular, toric varieties have proven useful both theoretically and in practical applications. A rich theory blending algebraic geometry and polyhedral geometry has been developed for T-varieties which are natural generalizations of toric varieties. The first results discussed in this dissertation study the relationship between torus actions and the well-poised property. In particular, I show that the well-poised property is preserved under a geometric invariant theory quotient by a (quasi-)torus. Conversely, I argue that T-varieties built on a well-poised base preserve the well-poised property when the base satisfies certain degree conditions. The second half of this dissertation covers two projects in algebraic statistics. The first studies level-1 phylogenetic network models which model evolutionary phenomena that trees fail to capture such as horizontal gene transfer and hybridization. In particular, I found the quadratic invariants of the Cavendar-Farris-Neyman model for level-1 networks and conjecture these generate the corresponding ideal. In the final project, I study a class of statistical models known as binary hierarchical models. Hierarchical models are known to be log-linear; thus, the joint probability distributions of the random variables naturally lie on a toric variety. For many applications such as testing normality of the model and finding a maximum likelihood estimate, a H-description of the marginal polytope is needed to drastically speed up computations. Here I provide an alternative polytope isomorphic to the marginal polytope in the binary case. This polytope is known as the generalized cut polytope, and I compute H-descriptions for all binary hierarchical models whose underlying simplicial complex is pure and of codimension 1