Maximum Likelihood Estimation for Nets of Conics

TL;DR

This paper investigates maximum likelihood estimation for nets of conics viewed as linear covariance models, analyzing their geometric properties and calculating the associated likelihood degrees.

Contribution

It provides a geometric analysis of reciprocal surfaces of nets of conics and computes their maximum likelihood degrees, extending Wall's classification of nets of conics.

Findings

Reciprocal surfaces are projections from the Veronese surface.

The intersection with polar nets is characterized geometrically.

Maximum likelihood degrees are explicitly computed.

Abstract

We study the problem of maximum likelihood estimation for $3$-dimensional linear spaces of $3\times 3$ symmetric matrices from the point of view of algebraic statistics where we view these nets of conics as linear concentration or linear covariance models of Gaussian distributions on $\mathbb{R}^3$. In particular, we study the reciprocal surfaces of nets of conics which are rational surfaces in $\mathbb{P}^5$. We show that the reciprocal surfaces are projections from the Veronese surface and determine their intersection with the polar nets. This geometry explains the maximum likelihood degrees of these linear models. We compute the reciprocal maximum likelihood degrees. This work is based on Wall's classification of nets of conics from 1977.