AA3 – Next Generation Networks



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


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. This will allow us to obtain results pertaining to the statistical problem of parameter estimation for these networks from given data.


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, e.g. 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

  • 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

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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