Extremal conditional independence for H\"usler-Reiss distributions via modular functions

We study extremal conditional independence for H\"{u}sler-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 H\"{u}sler-Reiss and Gaussian models to introduce a multiinformation-inspired measure $m^{\text{HR}}$ for H\"{u}sler-Reiss distributions. For the second function, we consider an invariant $\sigma^2$ that is naturally associated to the H\"{u}sler-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 H\"usler-Reiss parameters and its relation with the Gaussian elliptope.