Families of polytopes with rational linear precision in higher dimensions

TL;DR

This paper introduces a new family of lattice polytopes with rational linear precision by linking them to multinomial staged tree models with rational MLE, exploring their geometric and combinatorial properties.

Contribution

It establishes a novel connection between polytopes with rational linear precision and log-linear models with rational MLE, and studies their combinatorial structures.

Findings

Polytopes with rational linear precision correspond to certain log-linear models with rational MLE.

The paper characterizes when multinomial staged tree models are log-linear.

It analyzes the interplay between the normal fan's primitive collections and the Horn matrix.

Abstract

In this article we introduce a new family of lattice polytopes with rational linear precision. For this purpose, we define a new class of discrete statistical models that we call multinomial staged tree models. We prove that these models have rational maximum likelihood estimators (MLE) and give a criterion for these models to be log-linear. Our main result is then obtained by applying Garcia-Puente and Sottile's theorem that establishes a correspondence between polytopes with rational linear precision and log-linear models with rational MLE. Throughout this article we also study the interplay between the primitive collections of the normal fan of a polytope with rational linear precision and the shape of the Horn matrix of its corresponding statistical model. Finally, we investigate lattice polytopes arising from toric multinomial staged tree models, in terms of the combinatorics of their tree representations.