Decomposing conditional independence ideals with hidden variables

TL;DR

This paper analyzes determinantal ideals related to conditional independence models with hidden variables, providing explicit decompositions, conditions for primality, and combinatorial classifications.

Contribution

It introduces explicit ideal decompositions, identifies when components are prime, and derives formulas for their dimensions and classification.

Findings

Explicit decompositions of determinantal ideals for hidden variable models

Conditions under which components are prime ideals

Formulas for the dimensions and classification of components

Abstract

We study a family of determinantal ideals whose decompositions encode the structural zeros in conditional independence models with hidden variables. We provide explicit decompositions of these ideals and, for certain subclasses of models, we show that this is a decomposition into radical ideals by displaying Gr\"obner bases for the components. We identify conditions under which the components are prime, and establish formulas for the dimensions of these prime ideals. Moreover, we show that the components in the decomposition can be grouped into equivalence classes defined by their combinatorial structure, and we derive a closed formula for the number of such classes.