Geometry of rational quasi-independence models as toric fiber products

TL;DR

This paper explores the geometric structure of quasi-independence models using toric fiber products, revealing how their algebraic properties relate to model inference and ML-degree, with a focus on lower-order models and their classifications.

Contribution

It introduces the coordinate toric fiber product (cTFP) framework for quasi-independence models and characterizes when these models are cTFPs of simpler models, including classification of ML-degree 1 cases.

Findings

ML-degree 1 models are realizable as iterated toric fiber products of linear ideals.

Characterization of cTFPs among Lawrence lifts of 2-way models.

Necessary conditions for k-way models to have ML-degree 1 based on facial submodels.

Abstract

We investigate the geometry of a family of log-linear statistical models called quasi-independence models. The toric fiber product is useful for understanding the geometry of parameter inference in these models because the maximum likelihood degree is multiplicative under the TFP. We define the coordinate toric fiber product, or cTFP, and give necessary and sufficient conditions under which a quasi-independence model is a cTFP of lower-order models. We show that the vanishing ideal of every 2-way quasi-independence model with ML-degree 1 can be realized as an iterated toric fiber product of linear ideals. We also classify which Lawrence lifts of 2-way quasi-independence models are cTFPs and give a necessary condition under which a $k$-way model has ML-degree 1 using its facial submodels.