# Conditional independence ideals with hidden variables

## TL;DR

This paper investigates determinantal ideals associated with conditional independence statements involving hidden variables, highlighting a specific example where minimal primes are determinantal ideals, unlike the general case.

## Contribution

It introduces a generalized example of CI ideals with hidden variables where minimal primes are determinantal, expanding understanding of their algebraic structure.

## Key findings

- Minimal primes are determinantal ideals in the example studied.
- General case does not always have determinantal minimal primes.
- Provides algebraic insights into CI statements with hidden variables.

## Abstract

We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We focus on an example that generalizes the CI ideals of the intersection axiom. In this example, the minimal primes are again determinantal ideals, which is not true in general.