The Maximum Likelihood Data Singular Locus

TL;DR

This paper investigates the set of data points for which likelihood equations have solutions within a model's singular locus, extending understanding of ML-degree behavior in algebraic statistics.

Contribution

It characterizes the data singular locus where likelihood equations intersect the model's singularities, providing new insights into the algebraic structure of likelihood solutions.

Findings

Identifies the data locus where likelihood solutions lie in the singular part of the model.

Provides algebraic descriptions of the data singular locus.

Enhances understanding of the ML-degree in relation to model singularities.

Abstract

For general data, the number of complex solutions to the likelihood equations is constant and this number is called the (maximum likelihood) ML-degree of the model. In this article, we describe the special locus of data for which the likelihood equations have a solution in the model's singular locus.