Max-linear graphical models with heavy-tailed factors on trees of transitive tournaments

TL;DR

This paper studies max-linear graphical models with heavy-tailed factors on specialized tree structures of transitive tournaments, analyzing tail dependencies, model limits, and parameter identifiability in latent variable scenarios.

Contribution

It introduces a new class of max-linear models on trees of transitive tournaments, characterizes their tail behavior, and provides conditions for parameter identifiability with latent variables.

Findings

Limiting tail distribution involves independent increments along shortest trails.

Model mimics Markov random field behavior in tail dependencies.

Parameters are identifiable under a simple criterion related to latent nodes.

Abstract

Graphical models with heavy-tailed factors can be used to model extremal dependence or causality between extreme events. In a Bayesian network, variables are recursively defined in terms of their parents according to a directed acyclic graph (DAG). We focus on max-linear graphical models with respect to a special type of graphs, which we call a tree of transitive tournaments. The latter are block graphs combining in a tree-like structure a finite number of transitive tournaments, each of which is a DAG in which every two nodes are connected. We study the limit of the joint tails of the max-linear model conditionally on the event that a given variable exceeds a high threshold. Under a suitable condition, the limiting distribution involves the factorization into independent increments along the shortest trail between two variables, thereby imitating the behavior of a Markov random field. We are also interested in the identifiability of the model parameters in case some variables are latent and only a subvector is observed. It turns out that the parameters are identifiable under a criterion on the nodes carrying the latent variables which is easy and quick to check.