Algebraic Statistics

Algebraic statistics is the area of mathematics concerned with using tools from algebraic geometry, commutative algebra and combinatorics to answer questions about probability theory, statistics and their applications. In this way, research questions in statistics can benefit from existing algebraic tools while at the same time motivating the development of new algebraic concepts, algorithms and software.  Results derived in the context of algebraic statistics have found application in different areas, for example in biology in the context of learning gene regulatory networks from cross-sectional data as well as in the context of phylogenetics, where evolutionary relationships are encoded by polynomial constraints. Further applications include causal discovery and goodness of fit testing. Algebraic statistics is a rapidly evolving field where new connections continue to be made, both with other areas of mathematics as well as different areas of application. The goal of this mini symposium is to provide an overview of different current research within algebraic statistics ranging from the use of tropical geometry in statistics to identifiability in discrete Lyapunov models.

Janike Oldekop: Euler Stratifications of Second Hypersimplices via Delta-matroids

In algebraic statistics, Euler stratifications arise in the study of maximum likelihood (ML) degrees of toric varieties. This invariant serves as an algebraic complexity measure of ML estimation and depends on the embedding of the variety. For generic embeddings, the ML degree of a toric variety coincides with its degree, whereas non-generic embeddings give rise to an ML degree drop. The locus of such non-generic embeddings is described by the principal A-determinant. In this talk, we explain how the principal A-determinant of second hypersimplices is related to delta-matroids, and how the associated delta-matroid determines the ML degrees of toric varieties arising from second hypersimplices of small order.

Niharika Chakrabarty Paul: Polyhedral Geometry of Degree-Bounded CIM Polytopes

Characteristic imset polytopes are convex hulls of a collection of characteristic imset vectors which are 0/1 vectors that encode the Markov equivalence class of a directed acyclic graph. These polytopes are the feasible region of linear programs which correspond to causal discovery problems. We introduce a new family of these polytopes whose vertices correspond to directed acyclic graphs with bounded in-degree k. This yields a polytope whose dimension is only polynomial in the number of nodes of the underlying graphs instead of exponential. We show that when k = 1, this polytope is a matroid independence polytope and provide an extensive list of inequalities for k = 2 which we conjecture form a complete H-representation of the polytope. This is joint work with Jane Ivy Coons and Ben Hollering.

Rosa Preiss: Toric Varieties from Multiple Zeta Values

Given any graded polynomial ring over an algebraically closed field, we define the projective toric variety cut out by relations between monomials of a fixed degree. We specifically study the case of the free commutative algebra generated by Lyndon words in letters 2 and 3, where a Lyndon word is assigned the weight (degree) given by the sum of its digits, e.g. wt(23233)=13. Due to a result by Francis Brown, it is known that the algebra of multiple Zeta values, a generalisation of positive integer Riemann Zeta values, is generated by the values for precisely the Lyndon words in 2 and 3. Conjecturally, it is isomorphic to the polynomial ring in these Lyndon words. Thus, our toric ideals describe all the expected nonlinear relations between multiple zeta values of a fixed weight

Joint work with Annika Burmester, Steven Charlton, Abhiram Kidambi and Felix Lotter