Algebraic Study of Discrete Imsetal Models
Amira Alkeswani
TL;DR
This work studies conditional independence statements among discrete random variables through the imset framework, connecting combinatorial CI relations with toric ideals and the geometry of imset cones. By examining elementary and non-elementary CI relations for $n=3,4$ (including binary and mixed state spaces), it classifies CI relations via Markov and Graver bases, analyzes the associated CI ideals, and explores imsetal models from the faces of the elementary imset cone. Key findings include the identification of quadratic binomials as generators of non-elementary CI relations, a Segre-type structure for certain CI ideals, and a conjectural unifying binomial core $I_E$ governing primary decompositions across imsetal models. The results illuminate a deep link between geometric, algebraic, and combinatorial aspects of CI statements with potential for generalized, dimension-wide theory and improved computational methods in algebraic statistics.
Abstract
The method of imsets, introduced by Studený, provides a geometric and combinatorial description of conditional independence statements. Elementary conditional independence statements over a finite set of discrete random variables correspond to column vectors of a matrix generating a polyhedral cone, and the associated toric ideals encode algebraic relations among these statements. In this paper, we study discrete probability distributions on sets of three and four random variables, including both binary variables and combinations of binary and ternary variables. We investigate the structure of conditional independence ideals arising from elementary and non-elementary CI relations and analyze the algebraic properties of imsetal models induced by faces of the elementary imset cone. Our results highlight connections between combinatorial CI relations, their associated ideals, and the geometry of imset cones.