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Inference for max-linear Bayesian networks with noise

Mark Adams, Kamillo Ferry, Ruriko Yoshida

TL;DR

This work extends max-linear Bayesian networks (MLBNs) to settings with multiplicative noise on a known DAG and analyzes parameter estimation in log-space via tropical geometry. It develops two complementary inference approaches: (i) a Gaussian Mixture Model (GMM) framework with the EM algorithm to estimate edge-log weights $\omega_{ij}$, typically taking $\hat{\omega}_{ij} = \min_k \{\mu_k\}$, and (ii) a geometric, polytrope-based method that fits hyperplane boundaries to recover $\omega_{ij}$ through a quadratic program. The study reveals that the GMM estimator can be inconsistent under tail dependency or when edge-specific observations are scarce, while the tropical/hyperplane method remains robust in these regimes; a key finding is a threshold on edge-specific sample size below which GMM estimators fail. The results provide practical guidance for causal inference in extreme-value settings and lay groundwork for further tropical-estimation theory and extensions to more complex graph structures. Overall, the paper contributes two principled estimation pathways for MLBNs with noise, advancing reliable inference for extreme-event causality."

Abstract

Max-Linear Bayesian Networks (MLBNs) provide a powerful framework for causal inference in extreme-value settings; we consider MLBNs with noise parameters with a given topology in terms of the max-plus algebra by taking its logarithm. Then, we show that an estimator of a parameter for each edge in a directed acyclic graph (DAG) is distributed normally. We end this paper with computational experiments with the expectation and maximization (EM) algorithm and quadratic optimization.

Inference for max-linear Bayesian networks with noise

TL;DR

This work extends max-linear Bayesian networks (MLBNs) to settings with multiplicative noise on a known DAG and analyzes parameter estimation in log-space via tropical geometry. It develops two complementary inference approaches: (i) a Gaussian Mixture Model (GMM) framework with the EM algorithm to estimate edge-log weights , typically taking , and (ii) a geometric, polytrope-based method that fits hyperplane boundaries to recover through a quadratic program. The study reveals that the GMM estimator can be inconsistent under tail dependency or when edge-specific observations are scarce, while the tropical/hyperplane method remains robust in these regimes; a key finding is a threshold on edge-specific sample size below which GMM estimators fail. The results provide practical guidance for causal inference in extreme-value settings and lay groundwork for further tropical-estimation theory and extensions to more complex graph structures. Overall, the paper contributes two principled estimation pathways for MLBNs with noise, advancing reliable inference for extreme-event causality."

Abstract

Max-Linear Bayesian Networks (MLBNs) provide a powerful framework for causal inference in extreme-value settings; we consider MLBNs with noise parameters with a given topology in terms of the max-plus algebra by taking its logarithm. Then, we show that an estimator of a parameter for each edge in a directed acyclic graph (DAG) is distributed normally. We end this paper with computational experiments with the expectation and maximization (EM) algorithm and quadratic optimization.
Paper Structure (23 sections, 3 theorems, 31 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 23 sections, 3 theorems, 31 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Motivation
  4. Contributions of this paper
  5. Organization of this paper
  6. Preliminaries
  7. Graph Terminology
  8. Probability primer
  9. Bayesian Networks
  10. Tropical semirings and polytropes
  11. Recursive structural equations and max-linear Bayesian networks
  12. Parameter estimation under uncertainty
  13. Gaussian Mixture Models
  14. Geometric Estimation
  15. Computational Experiments
  16. ...and 8 more sections

Key Result

Lemma 2.5

Let $i\neq j\in V(G)$ be distinct nodes of the underlying graph $G$. Then, the random variable $Y_{ij}$ has an atom at $\omega_{kj} - \omega_{ki}$ for every common ancestor $k$ and these are the only atoms. In particular, if $i$ is an ancestor of $j$, then there is an atom at $\omega_{ij}$.

Figures (10)

  • Figure 1: A DAG consisting of four vertices. Each vertex $i$ in the network represents a random variable $X_i$ in the joint distribution of a random vector $X=(X_1, X_2, X_3, X_4)$.
  • Figure 2: This curated network provides the basis for our analysis throughout the paper. The unique combinations of directed edges creates triangles, diamonds, and max-weighted paths that create challenges in parameter estimation.
  • Figure 3: Graph structures our network integrates to examine parameter estimation in combination and isolation.
  • Figure 4: A max-linear Bayesian network on 4 nodes with $N = 2000$ along with the density plot of $Y_{24}$ and the marginal plot of $Y_{14}$ vs. $Y_{24}$. These visualizations provide insights into the effectiveness of our methodology, naming the application of GMM and the geometric of the associated polytrope.
  • Figure 5: Density plot of $Y_{24}$ from Example \ref{['ex:GMM']} together with uncertainty of assigning observations to one of the $K$ components of the Gaussian mixture. Assignment uncertainty is maximized at the meeting of two mixtures.
  • ...and 5 more figures

Theorems & Definitions (17)

  • Example 1.1
  • Remark 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Lemma 2.5: GKL:2019
  • Remark 2.6
  • Definition 3.1
  • Theorem 3.2
  • proof
  • ...and 7 more