Decomposing Conditional Independence Ideals with Hidden Variables: A Matroid-Theoretic Approach

2025/10/14 by Liwski, Emiliano
#Algebraic Geometry (math.AG) #Combinatorics (math.CO) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.2510.12570

Abstract

We study a class of determinantal ideals arising from conditional independence (CI) statements with hidden variables. Such CI statements translate into determinantal conditions on a matrix whose entries represent the probabilities of events involving the observed random variables. Our main objective is to determine the irreducible components of the corresponding varieties and to provide a combinatorial or geometric interpretation of each. We achieve this by introducing a new approach rooted in matroid theory. In particular, we introduce a new class of matroids, which we term quasi-paving matroids, and show that the components of these determinantal varieties are precisely matroid varieties of quasi-paving matroids. Moreover, we derive generating functions that encode the number of irreducible components of these CI ideals.

Citations

• Algebraic Geometry of Cactus, Pascal, and Pappus Matroids
• Decomposing conditional independence ideals with hidden variables
• Efficient Algorithms for Maximal Matroid Degenerations and Irreducible Decompositions of Circuit Varieties
• Solvable and Nilpotent Matroids: Realizability and Irreducible Decomposition of Their Associated Varieties
• Paving Matroids: Defining Equations and Associated Varieties
• Liftable Point-Line Configurations: Defining Equations and Irreducibility of Associated Matroid and Circuit Varieties
• Singular matroid realization spaces
• Matroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models
• Geometric Equations for Matroid Varieties
• On the Primary Decomposition of Some Determinantal Hyperedge Ideal
• Information-theoretic inference of common ancestors
• Binomial edge ideals and conditional independence statements
• Ideals of Adjacent Minors
• Groebner bases and determinantal ideals

Related

• Decomposing conditional independence ideals with hidden variables
• Matroid stratifications of hypergraph varieties, their realization spaces, and discrete conditional independence models
• An axiomatization of matroids and oriented matroids as conditional independence models
• On the Primary Decomposition of Some Determinantal Hyperedge Ideal
• Algebraic Aspects of Conditional Independence and Graphical Models