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Polyhedral Aspects of Maxoids

Tobias Boege, Kamillo Ferry, Benjamin Hollering, Francesco Nowell

TL;DR

The paper studies maxoids, the CI structures of max-linear Bayesian networks where the $C^*$-separation criterion depends on the edge-weight matrix $C$. It reveals a rich polyhedral structure: the set of weight matrices yielding a fixed maxoid forms full-dimensional cones that assemble into a complete fan, whose maximal cones correspond to generic CI structures; it further shows that the maxoid fan is the normal fan of a maxoid polytope built from path polynomials. By linking tropical geometry and polyhedral geometry, the authors provide an algorithmic approach to CI implication via polyhedral feasibility and SMT solvers, and they explore fundamental properties like compositional graphoid behavior, Amalgamation, and Strong Spohn, distinguishing maxoids from Gaussian models. They also introduce Maxoids.jl as a practical tool and discuss the broader implications for causal discovery in extreme-event settings.

Abstract

The conditional independence (CI) relation of a distribution in a max-linear Bayesian network depends on its weight matrix through the $C^\ast$-separation criterion. These CI~models, which we call maxoids, are compositional graphoids which are in general not representable by Gaussian random variables. We prove that every maxoid can be obtained from a transitively closed weighted DAG and show that the stratification of generic weight matrices by their maxoids yields a polyhedral~fan. We also use this connection to polyhedral geometry to develop an algorithm for solving the conditional independence implication problem for maxoids.

Polyhedral Aspects of Maxoids

TL;DR

The paper studies maxoids, the CI structures of max-linear Bayesian networks where the -separation criterion depends on the edge-weight matrix . It reveals a rich polyhedral structure: the set of weight matrices yielding a fixed maxoid forms full-dimensional cones that assemble into a complete fan, whose maximal cones correspond to generic CI structures; it further shows that the maxoid fan is the normal fan of a maxoid polytope built from path polynomials. By linking tropical geometry and polyhedral geometry, the authors provide an algorithmic approach to CI implication via polyhedral feasibility and SMT solvers, and they explore fundamental properties like compositional graphoid behavior, Amalgamation, and Strong Spohn, distinguishing maxoids from Gaussian models. They also introduce Maxoids.jl as a practical tool and discuss the broader implications for causal discovery in extreme-event settings.

Abstract

The conditional independence (CI) relation of a distribution in a max-linear Bayesian network depends on its weight matrix through the -separation criterion. These CI~models, which we call maxoids, are compositional graphoids which are in general not representable by Gaussian random variables. We prove that every maxoid can be obtained from a transitively closed weighted DAG and show that the stratification of generic weight matrices by their maxoids yields a polyhedral~fan. We also use this connection to polyhedral geometry to develop an algorithm for solving the conditional independence implication problem for maxoids.
Paper Structure (9 sections, 21 theorems, 56 equations, 7 figures, 2 tables)

This paper contains 9 sections, 21 theorems, 56 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The Polyhedral Geometry of C-star-separation
  3. The Maxoid Polytope
  4. Conditional Independence Implication for Maxoids
  5. Polyhedral and Tropical Geometry
  6. Cones and Fans
  7. Polytopes and normal fans
  8. Polynomials and Tropical Hypersurfaces
  9. Gröbner bases and fans

Key Result

Theorem 1

For any DAG $\mathcal{G}$ there is a hyperplane arrangement $\mathcal{H}_\mathcal{G} \subseteq \mathbb{R}^E$ such that for every $C \in \mathbb{R}^E \setminus \mathcal{H}_\mathcal{G}$ the set is a full-dimensional open polyhedral cone. The collection of all closures of such cones for a fixed $\mathcal{G}$ forms a complete polyhedral fan $\mathcal{F}_\mathcal{G}$ in $\mathbb{R}^E$. Moreover the ma

Figures (7)

  • Figure 1: The types of $\ast$-connecting paths between $i$ and $j$ given $L$ in a critical DAG $\mathcal{G}^{\ast}_{C,L}$. The colored colliders $\ell$ must belong to $L$; the non-colliders $p,q$ must not belong to $L$.
  • Figure 2: For appropriate $C^{\mathrm{tr}}$ and $\overline{C}$, $\mathcal{M}_\ast(\mathcal{G}, C) = \mathcal{M}_\ast(\mathcal{G}^{\mathrm{tr}}_C, C^{\mathrm{tr}}) = \mathcal{M}_\ast(\overline{\mathcal{G}}, \overline{C})$ holds.
  • Figure 3: The maxoid fan $\mathcal{F}_\mathcal{G}$ of the diamond, viewed in a 2-dimensional projection of $\mathbb{R}^4$. The full-dimensional cones in the fan correspond to the unique choices of ${\pi}^{1 4}_{\text{crit}}$ viewed on the right. The non-generic maxoid $\mathcal{M}_3$ is realized when the weights of the two paths coincide.
  • Figure 4: A complete weighted DAG on 3 nodes.
  • Figure 5: The maxoid polytope for the complete graph in \ref{['eg:Complete4']}. The colored vertex labels refer to the types of maxoids in \ref{['fig:hasse-diagram-maxoids']}.
  • ...and 2 more figures

Theorems & Definitions (57)

  • Theorem
  • Definition 2.1
  • Theorem 2.2: MaxLinearCI
  • Example 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 47 more