The Maximum Likelihood Degree

TL;DR

This paper explores the algebraic complexity of maximum likelihood estimation by analyzing the degree of critical equations, linking it to geometric and topological properties, and providing explicit formulas for generic cases.

Contribution

It establishes a connection between the maximum likelihood degree and topological invariants, offering explicit formulas for polynomials with generic coefficients.

Findings

Maximum likelihood degree relates to the number of bounded regions in hypersurface arrangements.

The degree equals the top Chern class of a sheaf of logarithmic differential forms under certain conditions.

Explicit formulas are derived for polynomials with generic coefficients.

Abstract

Maximum likelihood estimation in statistics leads to the problem of maximizing a product of powers of polynomials. We study the algebraic degree of the critical equations of this optimization problem. This degree is related to the number of bounded regions in the corresponding arrangement of hypersurfaces, and to the Euler characteristic of the complexified complement. Under suitable hypotheses, the maximum likelihood degree equals the top Chern class of a sheaf of logarithmic differential forms. Exact formulae in terms of degrees and Newton polytopes are given for polynomials with generic coefficients.