Motivated by extreme value theory, max-linear Bayesian networks have been recently introduced and studied as an alternative to linear structural equation models. However, for max-linear systems the classical independence results for Bayesian networks are far from exhausting valid conditional independence statements. We use tropical linear algebra to derive a compact representation of the conditional distribution given a partial observation, and exploit this to obtain a complete description of all conditional independence relations. In the context-specific case, where conditional independence is queried relative to a specific value of the conditioning variables, we introduce the notion of a source DAG to disclose the valid conditional independence relations. In the context-free case, we characterize conditional independence through a modified separation concept, ∗-separation, combined with a tropical eigenvalue condition. We also introduce the notion of an impact graph, which describes how extreme events spread deterministically through the network and we give a complete characterization of such impact graphs. Our analysis opens up several interesting questions concerning conditional independence and tropical geometry.
Carlos Améndola was partially supported by the Deutsche Forschungsgemeinschaft (DFG) in the context of the Emmy Noether junior research group KR 4512/1-1. Steffen Lauritzen has benefited from financial support from the Alexander von Humboldt Stiftung. Ngoc Tran would like to thank the Hausdorff Center for Mathematics for making her summer research visits to Germany possible.
Carlos Améndola. Claudia Klüppelberg. Steffen Lauritzen. Ngoc M. Tran. "Conditional independence in max-linear Bayesian networks." Ann. Appl. Probab. 32 (1) 1 - 45, February 2022. https://doi.org/10.1214/21-AAP1670