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Marginal Independence and Partial Set Partitions

Francisco Ponce-Carrión, Seth Sullivant

TL;DR

The paper develops a general combinatorial description of marginal independence by mapping statements to split closed order ideals in the poset $P\Pi_{n,2}$, establishing a bijection between marginal independence structures on $n$ variables and these ideals. In the discrete setting, the corresponding varieties are toric in $cdf$ coordinates, with a homogeneous parametrization linked via Möbius inversion on the index poset. An axiomatic framework with Decomposition, Join, and Splitting is shown to be sound and complete for all probability distributions, enabling a polynomial-time implication test. By unifying marginal independence notions beyond graphs and simplicial complexes, the work provides both theoretical insights and practical tools for constructing and interrogating marginal independence structures within algebraic statistics.

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.

Marginal Independence and Partial Set Partitions

TL;DR

The paper develops a general combinatorial description of marginal independence by mapping statements to split closed order ideals in the poset , establishing a bijection between marginal independence structures on variables and these ideals. In the discrete setting, the corresponding varieties are toric in coordinates, with a homogeneous parametrization linked via Möbius inversion on the index poset. An axiomatic framework with Decomposition, Join, and Splitting is shown to be sound and complete for all probability distributions, enabling a polynomial-time implication test. By unifying marginal independence notions beyond graphs and simplicial complexes, the work provides both theoretical insights and practical tools for constructing and interrogating marginal independence structures within algebraic statistics.

Abstract

We establish a bijection between marginal independence models on 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.
Paper Structure (4 sections, 23 theorems, 74 equations, 3 figures)

This paper contains 4 sections, 23 theorems, 74 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Marginal Independence and the Poset of Partial Set Partitions
  3. Axiomatic characterization of independence
  4. Discrete marginal independence models

Key Result

Theorem 1

Marginal independence models on $n$ random variables are in bijective correspondence to split closed order ideals in $P\Pi_{n,2}$.

Figures (3)

  • Figure 1: Hasse diagrams of $P\Pi_{3}$ and $P\Pi_{3,2}$
  • Figure 2: The order ideal generated by $1|23$ and $2|3$ from Example \ref{['ex:split']}
  • Figure 3: Index poset $\mathcal{P}$ from Example \ref{['ex:poset']}.

Theorems & Definitions (83)

  • Theorem
  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Proposition 2.7
  • ...and 73 more