  We study Bayesian networks based on max-linear structural equations as
introduced in Gissibl and Kl\"uppelberg [16] and provide a summary of their
independence properties. In particular we emphasize that distributions for such
networks are generally not faithful to the independence model determined by
their associated directed acyclic graph. In addition, we consider some of the
basic issues of estimation and discuss generalized maximum likelihood
estimation of the coefficients, using the concept of a generalized likelihood
ratio for non-dominated families as introduced by Kiefer and Wolfowitz [21].
Finally we argue that the structure of a minimal network asymptotically can be
identified completely from observational data.
