2020/02/29 by Carlos Améndola, Claudia Klüppelberg, Steffen L. Lauritzen +3 · 1 citation
Computer Science · Mathematics · #Algorithm #Bayesian Modeling and Causal Inference #Bayesian network #Conditional expectation #Conditional independence #Conditional probability #Conditional probability distribution #Context (archaeology) #Directed acyclic graph #Econometrics #Independence (probability theory) #Markov Chains and Monte Carlo Methods #Mathematics #Polynomial and algebraic computation #Representation (politics) #Statistics #math.PR #math.ST #msc:14T90 #msc:60G70 #msc:62H22 #msc:62R01 #stat.TH
paper · pdf · doi:10.1214/21-aap1670
published as Annals of Applied Probability 32, 1-45, 2022
arxiv created 2020/09/06 · openalex publication_date 2022/02/01 · arxiv updated 2022/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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.