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Bibliographic Details
Main Author:	Ferry, Kamillo
Format:	Recurso digital
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Published:	Zenodo 2026
Online Access:	https://doi.org/10.5281/zenodo.18119373
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Table of Contents:

Supplemental code and data for the PhD thesis "Algebra and Geometry of Max-Linear Bayesian Networks". Max-linear Bayesian networks are a type of statistical model introduced by Gissibl and Klüppelberg in 2018. Being graphical models they lean towards an intuitive presentation of (assumed) causal relationships, and as linear structural equation models over the tropical max-times semiring, they afford a reasonably intuitive interpretation of the encoded relationships. Yet, the tropical perspective on max-linear Bayesian networks has not been looked deeply into in the literature and thus holds untapped potential. This is a gap that this work aims to fill. As a starting point, we establish a connection between max-linear Bayesian networks and tropical polyhedra by using the linear structural equations attached to the former. In fact, the set of possible observations of a max-linear Bayesian network is a tropically and classically convex polyhedron, a polytrope. We proceed to study the combinatorics of tropical polyhedra arising from max-linear Bayesian networks using different methods. We give a characterization in terms of regular central subdivisions of type $A_n$ root polytopes, and in terms of the initial ideals of certain toric ideals. This also results in a polyhedral characterization of identifiability of network parameters for max-linear Bayesian networks. We also give a polyhedral classification of the conditional independence structure for a max-linear Bayesian network. This leads to the definition of the term maxoid in analogy to the notion of a matroid. We show that the different types of maxoids give a decomposition of the parameter space as a polyhedral fan, and compare the combinatorics of maxoids with the combinatorics of polytropes to find that maxoids give a coarser  classification than the associated polytropes. The structure of the set of possible observations as polytrope has consequences for parameter estimation of max-linear Bayesian networks. We use the geometry of the set to propose the minimum bounding polytrope of a sample as an estimator for the parameters of a network, and study properties such as the minimal size of a best-case sample for this estimator to recover all possible parameters. This is done using a combinatorial problem on the central subdivisions from before. While we study this estimator in an idealized, noise-free setting, we also use special distributional properties of max-linear Bayesian networks to study the problem of parameter estimation using noisy data. For this, we apply Gaussian mixtures and discuss methods for estimation of those.

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