Algebraic statistics of H\"usler-Reiss graphical models in multivariate extremes

TL;DR

This paper explores the algebraic geometric structure of H"usler-Reiss graphical models in multivariate extremes, focusing on polynomial constraints, maximum likelihood degree, and thresholds, revealing parallels and differences with Gaussian models.

Contribution

It introduces algebraic geometric methods to analyze extremal conditional independence in H"usler-Reiss models, including polynomial ideals and MLE properties.

Findings

Polynomial constraints characterize extremal CI relations.

Determinantal representation of extremal CI ideals.

Analysis of extremal MLE degree and thresholds.

Abstract

The field of extreme value statistics is concerned with modeling and predicting rare events. In a H\"usler-Reiss graphical model, a graph represents extremal conditional independence (CI) relations between random variables. These models are exponential families parameterized by a graph Laplacian and are considered the analogue of multivariate Gaussian models in the extremal setting. We study these models from the perspective of algebraic geometry. Translating the CI relations into polynomial constraints in the parameters, we define extremal CI ideals and find a determinantal representation of their generators. In terms of parametric inference, we study the extremal maximum likelihood degree as the number of solutions to a conditionally negative definite matrix completion problem. We also define and analyze the extremal maximum likelihood threshold for H\"usler-Reiss graphical models, which provides a certificate for the existence of a surrogate MLE in terms of the dimensionality of the point configuration that realizes the underlying summary statistic as a Euclidean distance matrix. We highlight throughout many interesting similarities but also differences with respect to Gaussian graphical models.