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Minimum bounding polytropes for estimation of max-linear Bayesian networks

Kamillo Ferry

TL;DR

This work advances identifiability and estimation for max-linear Bayesian networks (MLBNs) by translating the problem into tropical geometry. The authors show that after a log transform MLBN observations lie in a tropical polytrope, and they connect DAG weights to facets of a weighted digraph polyhedron $\mathrm{Q}(-\log C)$ via the Kleene star $C^\star$, enabling a geometric interpretation of the minimum-ratio estimator. They introduce minimum bounding polytropes as a practical device to certify parameter recovery and propose a set-cover framework to characterize minimal best-case samples needed for weight identification; they extend these ideas to structural learning with unknown DAGs by combining tropical geometry with scoring and thresholding techniques. Extensive experiments on simulated data and real-world datasets (NHANES 2015–2016 and the upper Danube network) assess the method’s performance, revealing strengths for moderate graphs and highlighting limitations due to noise and label ordering in large or complex networks. The work also raises open questions about the minimal set-cover sizes and the existence of closed forms for related combinatorial sequences, pointing to rich avenues for further investigation in tropical MLBNs.

Abstract

Max-linear Bayesian networks are recursive max-linear structural equation models represented by an edge weighted directed acyclic graph (DAG). The identifiability and estimation of max-linear Bayesian networks is an intricate issue as Gissibl, Klüppelberg, and Lauritzen have shown. As such, a max-linear Bayesian network is generally unidentifiable and standard likelihood theory cannot be applied. We can associate tropical polyhedra to max-linear Bayesian networks. Using this, we investigate the minimum-ratio estimator proposed by Gissibl, Klüppelberg, and Lauritzen and give insight on the structure of minimal best-case samples for parameter recovery which we describe in terms of set covers of certain triangulations. We also combine previous work on estimating max-linear models from Tran, Buck, and Klüppelberg to apply our geometric approach to the structural inference of max-linear models. This is tested extensively on simulated data and on real world data set, the NHANES report for 2015--2016 and the upper Danube network data.

Minimum bounding polytropes for estimation of max-linear Bayesian networks

TL;DR

This work advances identifiability and estimation for max-linear Bayesian networks (MLBNs) by translating the problem into tropical geometry. The authors show that after a log transform MLBN observations lie in a tropical polytrope, and they connect DAG weights to facets of a weighted digraph polyhedron via the Kleene star , enabling a geometric interpretation of the minimum-ratio estimator. They introduce minimum bounding polytropes as a practical device to certify parameter recovery and propose a set-cover framework to characterize minimal best-case samples needed for weight identification; they extend these ideas to structural learning with unknown DAGs by combining tropical geometry with scoring and thresholding techniques. Extensive experiments on simulated data and real-world datasets (NHANES 2015–2016 and the upper Danube network) assess the method’s performance, revealing strengths for moderate graphs and highlighting limitations due to noise and label ordering in large or complex networks. The work also raises open questions about the minimal set-cover sizes and the existence of closed forms for related combinatorial sequences, pointing to rich avenues for further investigation in tropical MLBNs.

Abstract

Max-linear Bayesian networks are recursive max-linear structural equation models represented by an edge weighted directed acyclic graph (DAG). The identifiability and estimation of max-linear Bayesian networks is an intricate issue as Gissibl, Klüppelberg, and Lauritzen have shown. As such, a max-linear Bayesian network is generally unidentifiable and standard likelihood theory cannot be applied. We can associate tropical polyhedra to max-linear Bayesian networks. Using this, we investigate the minimum-ratio estimator proposed by Gissibl, Klüppelberg, and Lauritzen and give insight on the structure of minimal best-case samples for parameter recovery which we describe in terms of set covers of certain triangulations. We also combine previous work on estimating max-linear models from Tran, Buck, and Klüppelberg to apply our geometric approach to the structural inference of max-linear models. This is tested extensively on simulated data and on real world data set, the NHANES report for 2015--2016 and the upper Danube network data.

Paper Structure

This paper contains 13 sections, 8 theorems, 30 equations, 8 figures, 6 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Tropical convexity
  4. Max-linear Bayesian models and data generation
  5. Regular subdivisions of root polytopes
  6. Calculating minimum bounding polytropes
  7. Parameter recovery and set covers
  8. Structural learning and gaps
  9. Experiments on data
  10. Simulated data
  11. Dietary supplement data
  12. The upper Danube network
  13. An open question

Key Result

Lemma 2.1

For any $C\in\mathbb{T}^{ \times d}$, if $C^\star$ converges, then $\mathrm{Q}(C) = \mathrm{Q}(C^\star)$.

Figures (8)

  • Figure 1: A central triangulation $\Sigma$ of the root polytope $P_{\kappa_4}$ and the link of the origin in $\Sigma$.
  • Figure 2: A schematic of \ref{['ex:underestimate-additional-facets']} where $P(S)$ exhibits more facets than $\mathrm{Q}(C)$. The additional facet of $P(S)$ is given by the dotted blue line.
  • Figure 3: Two examples of $\widetilde{\Sigma}$ for samples from $\mathrm{Q}(C)$ in \ref{['ex:dual-covering']}.
  • Figure 4: The maximal number of facets (blue circles) and pseudovertices (orange squares) of a polytrope $\mathrm{Q}(C)\subset\mathbb{T}\mathbb{A}^{d-1}$ when $C$ is supported on a DAG. Also shown is the minimal best case sample size (magenta triangles) when $C$ is supported on $\kappa_n$ (compare with \ref{['tab:enumeration']}).
  • Figure 5: Two central regular triangulations of $P_{\kappa_4}$ from \ref{['ex:dual-covering-4']} with a minimal set cover.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Lemma 2.1: Joswig:ETC
  • Definition 2.2
  • Definition 2.3
  • Theorem 2.4: AF:2024
  • Lemma 2.5
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Lemma 3.3
  • ...and 9 more