Estimating linear covariance models with numerical nonlinear algebra

TL;DR

This paper applies numerical nonlinear algebra techniques to improve maximum likelihood estimation in Gaussian covariance models with linear constraints, introducing new software and analyzing estimator properties.

Contribution

It introduces a novel application of numerical nonlinear algebra to covariance estimation and provides a software package for reliable computation of local maxima.

Findings

All local maxima can be computed reliably with the new software.

The paper identifies scenarios where the estimator is a rational function.

Analysis of maximum likelihood degree and its dual in constrained Gaussian models.

Abstract

Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that are of interest in statistics. We study the maximum likelihood degree and its dual analogue, and we introduce a new software package LinearCovarianceModels.jl for solving the score equations. All local maxima can thus be computed reliably. In addition we identify several scenarios for which the estimator is a rational function.