Computing Elimination Ideals and Discriminants of Likelihood Equations

TL;DR

This paper introduces a probabilistic algorithm that efficiently computes elimination ideals and discriminants of likelihood equations, significantly improving performance over traditional methods for large models in algebraic statistics.

Contribution

The paper presents a novel probabilistic approach leveraging theoretical insights into polynomial coefficients and Newton polytopes to efficiently compute elimination ideals and discriminants.

Findings

Successfully computed discriminants for large models like 3x3 matrix and Jukes-Cantor models

Achieved significant efficiency improvements over Groebner basis and interpolation methods

Handled large data files (over 30 GB) with practical computation times

Abstract

We develop a probabilistic algorithm for computing elimination ideals of likelihood equations, which is for larger models by far more efficient than directly computing Groebner bases or the interpolation method proposed in the first author's previous work. The efficiency is improved by a theoretical result showing that the sum of data variables appears in most coefficients of the generator polynomial of elimination ideal. Furthermore, applying the known structures of Newton polytopes of discriminants, we can also efficiently deduce discriminants of the elimination ideals. For instance, the discriminants of 3 by 3 matrix model and one Jukes-Cantor model in phylogenetics (with sizes over 30 GB and 8 GB text files, respectively) can be computed by our methods.