Max-linear Bayesian networks

We study Bayesian networks based on max-linear structural equations as introduced by Gissibl and Klüppelberg (2015) and provide a summary of their independence properties. In particular we emphasize that distributions for such networks are never faithful to the independence model determined by their associated directed acyclic graph unless the latter is a polytree, in which case they are always faithful. 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. Particular emphasis will be placed on the use of max-times algebra in the formulation and analysis of such models. The lecture is based on joint work with Gissibl and Klüppelberg.