Toric invariant theory for maximum likelihood estimation in log-linear models

TL;DR

This paper links invariant theory with maximum likelihood estimation in log-linear models, showing that norm minimization over a torus orbit corresponds to MLE and using stability concepts to determine MLE existence.

Contribution

It introduces a novel connection between invariant theory and statistical estimation, providing new criteria for MLE existence in log-linear models.

Findings

Norm minimization over a torus orbit equals MLE in log-linear models.

Stability under torus action characterizes MLE existence.

Discusses implications for scaling algorithms.

Abstract

We establish connections between invariant theory and maximum likelihood estimation for discrete statistical models. We show that norm minimization over a torus orbit is equivalent to maximum likelihood estimation in log-linear models. We use notions of stability under a torus action to characterize the existence of the maximum likelihood estimate, and discuss connections to scaling algorithms.