Marginal Independence and Partial Set Partitions

TL;DR

This paper establishes a mathematical framework linking marginal independence models with partial set partitions, showing they are toric and providing axiomatic characterizations for these models.

Contribution

It introduces a bijection between marginal independence models and split closed order ideals, generalizes prior results, and offers a complete axiomatic characterization.

Findings

Marginal independence models are toric in cdf coordinates.

A bijection exists between models and partial set partitions.

Axioms for marginal independence are sound and complete.

Abstract

We establish a bijection between marginal independence models on $n$ random variables and split closed order ideals in the poset of partial set partitions. We also establish that every discrete marginal independence model is toric in cdf coordinates. This generalizes results of Boege, Petrovic, and Sturmfels and Drton and Richardson, and provides a unified framework for discussing marginal independence models. Additionally, we provide an axiomatic characterization of marginal independence and we show that our set of axioms are sound and complete in the set of probability distributions. This follows the work of Geiger, Paz and Pearl who provided an analogous characterization of independence for statements involving 2 sets of random variables.