Extremal conditional independence for Hüsler-Reiss distributions via modular functions

We study extremal conditional independence for Hüsler-Reiss distributions, which is a parametric subclass of multivariate Pareto distributions. As the main contribution, we introduce two set functions, i.e.~functions which assign a value to the distribution and each of its marginals, and show that extremal conditional independence statements can be characterized by modularity relations for these functions. For the first function, we make use of the close connection between Hüsler-Reiss and Gaussian models to introduce a multiinformation-inspired measure $m^{\text{HR}}$ for Hüsler-Reiss distributions. For the second function, we consider an invariant $σ^2$ that is naturally associated to the Hüsler-Reiss parameterization and establish the second modularity criterion under additional positivity constraints. Together, these results provide new tools for describing extremal dependence structures in high-dimensional extreme value statistics. In addition, we study the geometry of a bounded subset of Hüsler-Reiss parameters and its relation with the Gaussian elliptope.

💡 Research Summary

This paper investigates extremal conditional independence (ECI) within the Hüsler‑Reiss (HR) family, a parametric subclass of multivariate Pareto distributions that serves as the extreme‑value analogue of multivariate Gaussian models. The authors introduce two set‑functions—(m^{\mathrm{HR}}) and (\sigma^{2})—that assign a scalar value to any subset of variables and to the full distribution, and they prove that modularity relations for these functions are exactly equivalent to ECI statements among arbitrary disjoint subsets of variables.

The first function, (m^{\mathrm{HR}}), is constructed by exploiting the close link between HR distributions and Gaussian models. For a subset (I\subseteq{1,\dots,d}) the function is defined as
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