Maximum information divergence from linear and toric models

TL;DR

This paper investigates the problem of maximizing information divergence from linear and toric models, introducing new geometric and algorithmic approaches, with special focus on reducible models and models with maximum likelihood degree one.

Contribution

It provides a geometric perspective using logarithmic Voronoi polytopes and develops an algorithm combining combinatorics and algebraic geometry for toric models.

Findings

Maximum divergence for linear models occurs at the boundary of the probability simplex.

An algorithm is proposed for toric models that leverages chamber complex combinatorics.

Special analysis is given for reducible models and models with maximum likelihood degree one.

Abstract

We study the problem of maximizing information divergence from a new perspective using logarithmic Voronoi polytopes. We show that for linear models, the maximum is always achieved at the boundary of the probability simplex. For toric models, we present an algorithm that combines the combinatorics of the chamber complex with numerical algebraic geometry. We pay special attention to reducible models and models of maximum likelihood degree one.