The maximum likelihood degree of sparse polynomial systems

TL;DR

This paper establishes that the maximum likelihood degree of sparse polynomial systems, which counts critical points of the likelihood function, is determined by the Newton polytopes and equals the mixed volume of a related Lagrange system.

Contribution

It provides a geometric formula linking the maximum likelihood degree to Newton polytopes and mixed volumes for sparse polynomial systems.

Findings

Maximum likelihood degree equals the mixed volume of a related Lagrange system.

The degree is determined by the Newton polytopes of the polynomial system.

Provides a geometric interpretation of critical points in statistical models.

Abstract

We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood degree counts the number of critical points of the likelihood function restricted to the model. We prove the maximum likelihood degree of a sparse polynomial system is determined by its Newton polytopes and equals the mixed volume of a related Lagrange system of equations.