Algebraic Study of Discrete Imsetal Models

Abstract

The method of imsets, introduced by Studen\'y, 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.