**Project Heads**

*Carlos Enrique Améndola Cerón*

**Project Members**

Kamillo Hugh Ferry

**Project Duration**

01.04.2023 – 31.03.2026

**Located at**

TU Berlin

**A lightning introduction**

The immediate question that arises from the connection between max-linear Bayesian networks and polytropes is to see if all possible combinatorial types arise from a max-linear Bayesian network or which actually do appear.

Above question is interesting in its own right, but it also serves as an intermediate step to the next goal. It is known that whether certain inequalities between the model parameters hold or not can lead to radically different conditional independence (CI) behavior on the same network. Thus, we aim to determine whether the combinatorics of the tropical polytope of a max-linear Bayesian network can reveal CI statements between subsets of variables, and viceversa.

Since max-linear Bayesian networks are a type of graphical model, one can also derive CI statements from the underlying graph structure. For other types of graphical models, like Gaussian networks, there exist separation but also algebraic criteria for conditional independence. Thus, we aim to find similar criteria in the max-linear Bayesian case.

**Project Webpages**

**Selected Publications
**

- Carlos Améndola and Kamillo Ferry. “
*Tropical combinatorics of weighted DAG polytropes*“. In preparation. 2024.

**Selected Pictures
**

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

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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