  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, $\ast$-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.
