Bounding the maximum likelihood degree

TL;DR

This paper establishes an upper bound for the maximum likelihood degree of algebraic statistical models using intersection-cohomology Euler characteristic, and provides counterexamples to a previously conjectured bound.

Contribution

It introduces a new bound for the maximum likelihood degree based on advanced algebraic topology and disproves a prior conjecture with counterexamples.

Findings

Maximum likelihood degree is bounded by the signed intersection-cohomology Euler characteristic.

Counterexamples show the usual Euler characteristic bound does not always hold.

The work connects algebraic statistics with topological invariants.

Abstract

Maximum likelihood estimation is a fundamental computational problem in statistics. In this note, we give a bound for the maximum likelihood degree of algebraic statistical models for discrete data. As usual, such models are identified with special very affine varieties. Using earlier work of Franecki and Kapranov, we prove that the maximum likelihood degree is always less or equal to the signed intersection-cohomology Euler characteristic. We construct counterexamples to a bound in terms of the usual Euler characteristic conjectured by Huh and Sturmfels.