Maximum likelihood degree of the two-dimensional linear Gaussian covariance model

TL;DR

This paper calculates the maximum likelihood degree for a generic two-dimensional subspace of Gaussian covariance matrices, revealing it to be 2n-3, which aids in algebraic statistical analysis.

Contribution

It provides a precise formula for the maximum likelihood degree of a specific Gaussian covariance model using intersection theory.

Findings

Maximum likelihood degree is 2n-3 for the model.

The result applies to generic two-dimensional subspaces.

The approach uses algebraic geometry techniques.

Abstract

In algebraic statistics, the maximum likelihood degree of a statistical model is the number of complex critical points of its log-likelihood function. A priori knowledge of this number is useful for applying techniques of numerical algebraic geometry to the maximum likelihood estimation problem. We compute the maximum likelihood degree of a generic two-dimensional subspace of the space of $n\times n$ Gaussian covariance matrices. We use the intersection theory of plane curves to show that this number is $2n-3$.