Minimum bounding polytropes for estimation of max-linear Bayesian networks

TL;DR

This paper explores the geometric structure of max-linear Bayesian networks using tropical polyhedra, providing new insights into parameter estimation and structural inference, supported by extensive simulations and real data applications.

Contribution

It introduces a geometric approach using minimum bounding polytropes for estimating max-linear Bayesian networks, enhancing understanding of their identifiability and estimation.

Findings

The geometric approach improves parameter recovery in max-linear models.

Minimal set covers relate to best-case samples for estimation.

Method performs well on simulated and real-world datasets.

Abstract

Max-linear Bayesian networks are recursive max-linear structural equation models represented by an edge weighted directed acyclic graph (DAG). The identifiability and estimation of max-linear Bayesian networks is an intricate issue as Gissibl, Kl\"uppelberg, and Lauritzen have shown. As such, a max-linear Bayesian network is generally unidentifiable and standard likelihood theory cannot be applied. We can associate tropical polyhedra to max-linear Bayesian networks. Using this, we investigate the minimum-ratio estimator proposed by Gissibl, Kl\"uppelberg, and Lauritzen and give insight on the structure of minimal best-case samples for parameter recovery which we describe in terms of set covers of certain triangulations. We also combine previous work on estimating max-linear models from Tran, Buck, and Kl\"uppelberg to apply our geometric approach to the structural inference of max-linear models. This is tested extensively on simulated data and on real world data set, the NHANES report for 2015--2016 and the upper Danube network data.