Extremes of Markov random fields on block graphs: max-stable limits and structured H\"usler-Reiss distributions

TL;DR

This paper investigates the extremal behavior of Markov random fields on block graphs, establishing max-stable limits and structured H"usler-Reiss distributions, with implications for multivariate extremes and extremal graphical models.

Contribution

It generalizes extremal results from Markov trees to block graphs, characterizes max-stable limits, and reveals structured H"usler-Reiss distributions in this context.

Findings

Limiting distribution determined by unique shortest paths in block graphs

Component-wise maxima attracted to structured max-stable H"usler-Reiss distributions

Parameters remain identifiable even with latent variables

Abstract

We study the joint occurrence of large values of a Markov random field or undirected graphical model associated to a block graph. On such graphs, containing trees as special cases, we aim to generalize recent results for extremes of Markov trees. Every pair of nodes in a block graph is connected by a unique shortest path. These paths are shown to determine the limiting distribution of the properly rescaled random field given that a fixed variable exceeds a high threshold. The latter limit relation implies that the random field is multivariate regularly varying and it determines the max-stable distribution to which component-wise maxima of independent random samples from the field are attracted. When the sub-vectors induced by the blocks have certain limits parametrized by H\"usler-Reiss distributions, the global Markov property of the original field induces a particular structure on the parameter matrix of the limiting max-stable H\"usler-Reiss distribution. The multivariate Pareto version of the latter turns out to be an extremal graphical model according to the original block graph. Thanks to these algebraic relations, the parameters are still identifiable even if some variables are latent.