Maximum Likelihood for Dual Varieties

TL;DR

This paper introduces an algebraic geometric approach to maximum likelihood estimation for discrete data models, reformulating the problem using dual varieties to simplify computations.

Contribution

It presents a novel reformulation of the MLE problem via dual and conormal varieties, enabling solutions without explicit model equations.

Findings

Dual likelihood equations are derived.

Solving the dual MLE problem provides solutions to the original MLE.

The approach simplifies MLE computation for algebraic statistical models.

Abstract

Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. In this paper, MLE for statistical models with discrete data is studied from an algebraic statistics viewpoint. A reformulation of the MLE problem in terms of dual varieties and conormal varieties will be given. With this description, the dual likelihood equations and the dual MLE problem are defined. We show that solving the dual MLE problem yields solutions to the MLE problem, so we can solve the MLE problem without ever determining the defining equations of the model.