Data-Discriminants of Likelihood Equations

TL;DR

This paper introduces data-discriminants (DD) to classify data based on the number of real solutions to likelihood equations in maximum likelihood estimation, using an efficient probabilistic algorithm.

Contribution

It develops a probabilistic algorithm to compute data-discriminants, enabling classification of data by real solutions in likelihood equations, improving efficiency over standard methods.

Findings

Probabilistic algorithm outperforms elimination algorithms in efficiency.

Data-discriminants effectively classify data by real solution counts.

Experimental results demonstrate practical applicability for benchmark models.

Abstract

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.