A PC Algorithm for Max-Linear Bayesian Networks

2025-08-19

AI Summary

Max-linear Bayesian networks (MLBNs) model heavy-tailed variables but violate d-separation faithfulness, rendering standard constraint-based causal discovery algorithms—such as PC—inapplicable. To address this, we propose a structure learning framework grounded in *-separation, the first to extend the PC algorithm to this non-standard separation criterion. We design PCstar, which leverages C*-separation to strengthen edge orientation, integrating conditional independence testing with constraint-based learning to consistently estimate the causal graph of MLBNs. We prove that PCstar is statistically consistent under *-separation. Empirical evaluations demonstrate that PCstar orients significantly more edges than the standard PC algorithm and achieves substantially higher structural recovery accuracy in heavy-tailed settings.

Technology Category

• Machine Learning: Causal Learning
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Max-linear Bayesian networks (MLBNs) are a relatively recent class of structural equation models which arise when the random variables involved have heavy-tailed distributions. Unlike most directed graphical models, MLBNs are typically not faithful to d-separation and thus classical causal discovery algorithms such as the PC algorithm or greedy equivalence search can not be used to accurately recover the true graph structure. In this paper, we begin the study of constraint-based discovery algorithms for MLBNs given an oracle for testing conditional independence in the true, unknown graph. We show that if the oracle is given by the $ast$-separation criteria in the true graph, then the PC algorithm remains consistent despite the presence of additional CI statements implied by $ast$-separation. We also introduce a new causal discovery algorithm named "PCstar" which assumes faithfulness to $C^ast$-separation and is able to orient additional edges which cannot be oriented with only d- or $ast$-separation.

Problem

• Develops PC algorithm for max-linear Bayesian networks
• Addresses unfaithfulness to d-separation in heavy-tailed distributions
• Introduces new algorithm for improved edge orientation

Innovation

• PC algorithm adapted for max-linear networks
• New PCstar algorithm using C*-separation criteria
• Constraint-based discovery with *-separation oracle