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.

Roan Talbut: Tropical Geometry for Phylogenetic Statistics

The identification of the tropical Grassmannian and the space of phylogenetic trees has inspired a range of research on the use of tropical geometry for phylogenetic statistics. We will review these connections and the various avenues of research which have followed. In particular, we consider the problem of comparing probability distributions on tropical spaces of differing dimensions. We construct a Wasserstein distance between measures on different tropical projective tori via tropical projections of probability measures, and show that this distance is symmetric, whether mapping from a low dimensional space to a high dimensional space or vice versa.

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

Nataliia Kushnerchuk: Posets of trek polynomials for directed graphical models

When a variety V𝜑 equals the image of a polynomial map 𝜑 whose coordinate functions are combinatorial generating polynomials (i.e. polynomials enumerating combinatorial objects), the geometry of V𝜑 reflects identities satisfied by the generating polynomials. The resulting interplay between combinatorics and algebraic geometry can be used to answer questions about V𝜑 . A recent technique proposes to do so using a partially ordered set (poset) 𝑃𝜑 defined via the coefficient vectors of the polynomials defining 𝜑. In this talk I am going to talk about a subfamily of Gaussian directed graphical models that we have studied with this new technique. For this family, the generating polynomials of 𝜑 defining a graphical model enumerate certain subgraphs known as treks. We characterized the poset 𝑃𝜑, and used the characterization to compute the linear span of V𝜑 , prove it is toric and deduce a basis for its vanishing ideal. As an additional consequence, it is shown that the varieties for two distinct directed trees intersect in a strictly lower-dimensional variety. This solves an instance of the structural identifiability problem in the graphical models program from statistics.

Based on https://arxiv.org/abs/2608.08325 

Lakshmi Ramesh: Maximum likelihood estimation of the location of a symmetric convex body

Consider data points sampled independently from the uniform distribution on a known symmetric convex body in high-dimensional Euclidean space with unknown location parameter. In this setting, the set of maximum likelihood estimators (MLE set) is a convex body containing the true location parameter. The goal of this talk is to present non-asymptotic upper and lower bounds for the diameter of the MLE set. 

Based on joint work with Vladimir Koltchinskii and Martin Wahl.