The maximum likelihood degree of mixtures of independence models

TL;DR

This paper investigates the algebraic complexity of maximum likelihood estimation for mixtures of independence models, providing recursive formulas and closed forms for their ML degree, which was previously unknown.

Contribution

It proves a conjecture by deriving recursions and explicit formulas for the ML degree of mixture models, advancing understanding of their algebraic properties.

Findings

Derived recursions for ML degree of mixture models

Provided closed-form expressions for ML degree

Confirmed conjecture on ML degree behavior

Abstract

The maximum likelihood degree (ML degree) measures the algebraic complexity of a fundamental optimization problem in statistics: maximum likelihood estimation. In this problem, one maximizes the likelihood function over a statistical model. The ML degree of a model is an upper bound to the number of local extrema of the likelihood function and can be expressed as a weighted sum of Euler characteristics. The independence model (i.e. rank one matrices over the probability simplex) is well known to have an ML degree of one, meaning their is a unique local maxima of the likelihood function. However, for mixtures of independence models (i.e. rank two matrices over the probability simplex), it was an open question as to how the ML degree behaved. In this paper, we use Euler characteristics to prove an outstanding conjecture by Hauenstein, the first author, and Sturmfels; we give recursions and closed form expressions for the ML degree of mixtures of independence models.