SF2704 Topics in Mathematics I 7.5 credits

Welcome to the course in Algebraic Statistics!

This course gives an introduction to the emerging field of algebraic statistics, which focuses on using methods from algebra to develop tools for statistical inference. In statistics, we typically start with a set of parameters that we use to define a distribution.  Oftentimes, the model-defining map that sends the parameter values to their distribution can be viewed a rational map.  From this perspective, the set of distributions we obtain from our space of possible parameter values (i.e. the statistical model) is the solution set to a collection of polynomial equations.  An understanding of these polynomial equations can then be used to develop statistical inference methods for solving fundamental problems related to point estimation, hypothesis testing, model selection and representation learning. To extract these statistical methods we will dig into the nature of these polynomial equations, utilizing methods from algebra, geometry and combinatorics.  Topics to be discussed include the geometry of discrete and Gaussian exponential families, the geometry of maximum likelihood inference, algebraic hypothesis tests for hierarchical models, parameter identifiability and the geometry of conditional independence models. Applications in categorical data analysis, causal inference and phylogenetics will be explored. 

Prerequisites

Knowledge obtained in a basic statistics course, linear algebra course, and discrete mathematics course is required. More advanced knowledge from courses in groups and rings would be helpful, but we will start the course with a soft introduction to the necessary tools from computational algebra. 

Course literature

Sullivant, Seth. Algebraic statistics. Vol. 194. American Mathematical Society, 2023.

Cox, David, et al. Ideals, varieties, and algorithms. Vol. 3. New York: Springer, 1997.