Tropical combinatorics of max-linear Bayesian networks

TL;DR

This paper explores the combinatorial structure of polytropes linked to max-linear Bayesian networks, revealing how network weights influence their geometric facets and aiding in understanding model identifiability.

Contribution

It provides a classification of polytropes from weighted DAGs and connects their structure to the statistical property of identifiability in max-linear Bayesian networks.

Findings

Edge weights determine the facet structure of associated polytropes.

Classification of polytropes based on weighted DAG equivalence.

Insights into the identifiability of max-linear Bayesian network models.

Abstract

A polytrope is a tropical polyhedron that is also classically convex. We study the tropical combinatorial types of polytropes associated to weighted directed acyclic graphs (DAGs). This family of polytropes arises in algebraic statistics when describing the model class of max-linear Bayesian networks. We show how the edge weights of a network directly relate to the facet structure of the corresponding polytrope. We also give a classification of polytropes from weighted DAGs at different levels of equivalence. These results give insight on the statistical problem of identifiability for a max-linear Bayesian network.