AA3 – Next Generation Networks

Project

AA3-16

Likelihood Geometry of Max-Linear Bayesian Networks

Project Heads

Carlos Enrique Améndola Cerón

Project Members

Kamillo Hugh Ferry

Project Duration

01.04.2023 – 31.03.2026

Located at

TU Berlin

Description

Max-linear Bayesian networks have emerged from the need to model cause and effect relations between large observed values of several variables. The structural equations that govern these relations via the model parameters can be reinterpreted in the language of tropical geometry. We will exploit this geometric connection to gain new insights on the possible combinatorial structures on the support of the random variables.

A lightning introduction

Above connection between our models and tropical geometry manifests itself in a tropical polytope that we can associate to the model parameters. The immediate goal is then to study the combinatorics of these tropical polytopes. While the combinatoric structure is rich, the setup of this question itself might even be explained in 5 minutes!

 

Results

There are two places where we could successfully investigate the properties of MLBNs from the perspective of tropical and polyhedral geometry. For one, we shed light on the identifiability of parameters in MLBNs using the connection to ordinarily convex tropical polyhedra (Amèndola and Ferry 2025). This also results in some enumerative results regarding triangulations of certain fundamental polytopes and groundwork for a moduli space for MLBNs.

This connection between tropical polyhedra and MLBNs also allows us to study a parameter estimator. Due to technical reasons, this minimum (tropical) ratio estimator as proposed by Gissibl, Klüppelberg and Lauritzen is not a maximum likelihood estimator in the classical sense. Yet we are able to connect the capability of this estimator to recover parameters to a combinatorial question (Ferry 2025).

As a next step, we investigated the structure of conditional independence (CI) statements, that are valid for MLBNs. It is already known that whether certain inequalities between the model parameters hold or not can lead to radically different CI behavior on the same network. It turns out that grouping MLBNs by their CI structures leads to a coarsening of the classification by tropical polytopes. Boege et al. (2025) described the necessary linear equations between the model parameters that distinguish between different CI structures.

Project Webpages

  • Directed separation in graphs is a small applet that demonstrates separation statements in directed graphs, which correspond to conditional independence in Bayesian networks.
A screenshot of a graphical software that displays a graph with five nodes.

Selected Publications

  • Kamillo Ferry. Minimum bounding polytropes for estimation of max-linear Bayesian networks. arXiv:2511.05962
  • Tobias Boege, Kamillo Ferry, Benjamin Hollering, Francesco Nowell. Polyhedral aspects of maxoids. In: Proceedings of the 13th Workshop on Uncertainty Processing. arXiv:2504.21068
  • Mark Adams, Kamillo Ferry, Ruriko Yoshida. Inference for Max-Linear Bayesian Network with noise. In: Proceedings of the 13th Workshop on Uncertainty Processing. arXiv:2505.00229
  • Carlos Améndola and Kamillo Ferry. Tropical combinatorics of max-linear Bayesian networks. Journal of Symbolic Computation (2025), 102518.

Selected Pictures

A diamond-shaped Bayesian network on the left with structural equations on the right. Diamond-shaped as in, a graph with one node that points towards two more nodes. These two nodes point toward a fourth common node. The equations read: X1 = Z1 X2 = max(c21X1, Z2) X3 = max(c31X1, Z3) X4 = max(c43X2, c43X3, Z4)

Max-Linear Bayesian Networks

These are graphical models with recursive structural equations in the max-times semiring. One example is given right above.

A small bayesian network consisting of five nodes. Nodes 1 and 2 point towards node 4 and nodes 2 and 3 point towards 5. 1 is the first marked subset, 4 and 5 the second marked subset and 3 the third marked subset. This image shows that 1 is star-separated from 3 if we condition on 4 and 5.

Separation statements in Bayesian networks

yield conditional independence statements. This means we can reason about the conditional independence of variables in our model by turning to reachability in the underlying graph. In above case, we can conclude that 1 is independent from 3 given 4 and 5, which means that knowing 1 and then finding out about 4 and 5 does not give any information about 3, at least in a max-linear Bayesian network.

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