A Probabilistic Algorithm for Computing Data-Discriminants of Likelihood Equations

TL;DR

This paper introduces a probabilistic algorithm to efficiently compute data-discriminants of likelihood equations, aiding in classifying parameters based on the number of real solutions in maximum likelihood estimation.

Contribution

It presents a novel probabilistic approach with multiple strategies for computing data-discriminants, improving efficiency over previous methods and standard elimination techniques.

Findings

The algorithm outperforms previous versions and standard methods in efficiency.

Application to a 3x3 symmetric matrix model demonstrates practical utility.

The approach enables effective real root classification in algebraic statistical models.

Abstract

An algebraic approach to the maximum likelihood estimation problem is to solve a very structured parameterized polynomial system called likelihood equations that have finitely many complex (real or non-real) solutions. The only solutions that are statistically meaningful are the real solutions with positive coordinates. In order to classify the parameters (data) according to the number of real/positive solutions, we study how to efficiently compute the discriminants, say data-discriminants (DD), of the likelihood equations. We develop a probabilistic algorithm with three different strategies for computing DDs. Our implemented probabilistic algorithm based on Maple and FGb is more efficient than our previous version presented in ISSAC2015, and is also more efficient than the standard elimination for larger benchmarks. By applying RAGlib to a DD we compute, we give the real root classification of 3 by 3 symmetric matrix model.