Marginal Independence Models

TL;DR

This paper studies marginal independence models defined by rank constraints on tensor marginals, exploring their algebraic structure, parameter estimation, and applications to graph models and matroids.

Contribution

It advances understanding of the toric ideals of these models and develops numerical methods for parameter estimation, with a comprehensive database of small models.

Findings

Toric ideals of marginal independence models are characterized.

Numerical algebraic methods for parameter estimation are developed.

A database of small models is presented.

Abstract

We impose rank one constraints on marginalizations of a tensor, given by a simplicial complex. Following work of Kirkup and Sullivant, such marginal independence models can be made toric by a linear change of coordinates. We study their toric ideals, with emphasis on random graph models and independent set polytopes of matroids. We develop the numerical algebra of parameter estimation, using both Euclidean distance and maximum likelihood, and we present a comprehensive database of small models.