Discrete Max-Linear Bayesian Networks

TL;DR

This paper introduces discrete max-linear Bayesian networks, extending max-linear models to discrete variables, and shows their algebraic structure is similar to conjunctive Bayesian networks, enabling the transfer of analytical techniques.

Contribution

It generalizes the algebraic properties of conjunctive Bayesian networks to a broader class of discrete max-linear models, including their toric variety structure.

Findings

Discrete max-linear Bayesian networks are isomorphic to CBN models when variables are binary.

Many analytical techniques from CBN models extend to discrete max-linear models.

Discrete max-linear models form toric varieties after linear coordinate transformation.

Abstract

Discrete max-linear Bayesian networks are directed graphical models specified by the same recursive structural equations as max-linear models but with discrete innovations. When all of the random variables in the model are binary, these models are isomorphic to the conjunctive Bayesian network (CBN) models of Beerenwinkel, Eriksson, and Sturmfels. Many of the techniques used to study CBN models can be extended to discrete max-linear models and similar results can be obtained. In particular, we extend the fact that CBN models are toric varieties after linear change of coordinates to all discrete max-linear models.