The binomial ideal of the intersection axiom for conditional probabilities

TL;DR

This paper proves that the binomial ideal related to the intersection axiom in conditional probability is radical and can be expressed as an intersection of toric prime ideals, resolving a conjecture in algebraic statistics.

Contribution

It establishes the radicality and prime ideal decomposition of the binomial ideal associated with the intersection axiom, confirming a conjecture in algebraic statistics.

Findings

The binomial ideal is radical.

It can be expressed as an intersection of toric prime ideals.

Resolves a conjecture by Cartwright and Engström.

Abstract

The binomial ideal associated with the intersection axiom of conditional probability is shown to be radical and is expressed as intersection of toric prime ideals. This resolves a conjecture in algebraic statistics due to Cartwright and Engstr\"om.