Likelihood Geometry

TL;DR

This paper explores the geometric and algebraic properties of likelihood functions over algebraic varieties, introducing new results on the likelihood correspondence and its bidegree, with applications in algebraic statistics.

Contribution

It provides an introduction to likelihood geometry, connecting it with combinatorial algebraic geometry objects and presenting new theoretical results on the likelihood correspondence.

Findings

Likelihood degree as a topological invariant

New results on likelihood correspondence and bidegree

Connections between algebraic geometry objects and statistical models

Abstract

We study the critical points of monomial functions over an algebraic subset of the probability simplex. The number of critical points on the Zariski closure is a topological invariant of that embedded projective variety, known as its maximum likelihood degree. We present an introduction to this theory and its statistical motivations. Many favorite objects from combinatorial algebraic geometry are featured: toric varieties, A-discriminants, hyperplane arrangements, Grassmannians, and determinantal varieties. Several new results are included, especially on the likelihood correspondence and its bidegree. These notes were written for the second author's lectures at the CIME-CIRM summer course on Combinatorial Algebraic Geometry at Levico Terme in June 2013.