# Bayesian Networks for Max-linear Models

## TL;DR

This paper explores Bayesian networks based on max-linear models, analyzing their independence properties, estimation methods, and the potential for structure identification from observational data.

## Contribution

It provides a comprehensive summary of independence properties, discusses generalized maximum likelihood estimation, and shows that minimal network structures can be identified asymptotically.

## Key findings

- Distributions are generally not faithful to the independence model.
- Generalized maximum likelihood estimation is applicable for coefficient estimation.
- Minimal network structures can be asymptotically identified from observational data.

## Abstract

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.