Algebraic and Holonomic Statistics

Lecture series at the Hypergeometric School 2023
Kobe University, August 16-18, 2023

Abstract:
This lecture series investigates the maximum likelihood estimation (MLE) problem from statistics through the lens of algebraic analysis and geometry. Maximum likelihood estimation is about determining parameters of a statistical model which probabilistically best explain observed data. Mathematically, the task is to maximize the likelihood function. In some cases, such as for the Fisher or Wishart distribution, this boils down to maximizing a holonomic function of the model parameters and can be carried out via the holonomic gradient ascent. An algebro-geometric study of the MLE problem for discrete statistical models is provided by likelihood geometry. Likelihood geometry encodes discrete statistical models as very affine varieties, i.e., as algebraic subvarieties of an algebraic torus. It turns out that there is a deep connection between the boundary components of a tropical compactification of the very affine variety and the codimension-one components of the Bernstein-Sato variety of a polynomial parameterization of the model, on which this lecture series sheds light.

Notes: Lecture 1, Lecture 2, Exercises

Main references:

  1. Michael F. Adamer, Andras Lőrincz, A.-L. S., and Bernd Sturmfels. Algebraic Analysis of Rotation Data. Algebraic Statistics, 11(2):189-211, 2020.
  2. June Huh and Bernd Sturmfels. Combinatorial Algebraic Geometry, volume 2108 of Lecture Notes in Mathematics, chapter Likelihood Geometry. Springer, Cham, 2014.
  3. Mutsumi Saito, Bernd Sturmfels, and Nobuki Takayama. Gröbner Deformations of Hypergeometric Differential Equations, volume 6 of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, 2000.
  4. A.-L. S. and Bernd Sturmfels. D-Modules and Holonomic Functions. In Varieties, polyhedra, computation. Volume 22 of EMS Series of Congress Reports, pp. 251-293, EMS Press, Berlin, 2025.
  5. A.-L. S. and Robin van der Veer. Maximum Likelihood Estimation from a Tropical and a Bernstein-Sato Perspective. International Mathematics Research Notices, 2023:5263-5292, 2023.