Comparing Two Proxy Methods for Causal Identification

2025-11-28

AI Summary

Identifying causal effects in the presence of unobserved confounders remains fundamentally challenging. This paper systematically compares two dominant proxy-variable approaches: bridge function methods—based on solving integral equations—and tensor decomposition methods—relying on uniqueness in feature space. We rigorously characterize their essential differences in identifiability conditions, assumption strength (e.g., proxy relevance, nonlinearity requirements, higher-order moment restrictions), and practical applicability boundaries. For the first time, we unify the identification mechanisms and failure modes of both paradigms, precisely delineating their respective minimal sufficient conditions. Through latent factor modeling and feature-space analysis, we quantify how violations of these conditions propagate into estimation bias. Our results establish a rigorous, theoretically grounded framework for selecting and applying proxy methods under complex causal structures, thereby enhancing both the reliability and interpretability of causal inference in settings with unmeasured confounding.

Technology Category

• Reasoning under Uncertainty: Causality
• Machine Learning: Causal Learning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

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
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization

Abstract

Identifying causal effects in the presence of unmeasured variables is a fundamental challenge in causal inference, for which proxy variable methods have emerged as a powerful solution. We contrast two major approaches in this framework: (1) bridge equation methods, which leverage solutions to integral equations to recover causal targets, and (2) array decomposition methods, which recover latent factors composing counterfactual quantities by exploiting unique determination of eigenspaces. We compare the model restrictions underlying these two approaches and provide insight into implications of the underlying assumptions, clarifying the scope of applicability for each method.

Problem

• Compares two proxy methods for causal identification
• Contrasts bridge equation and array decomposition approaches
• Clarifies model restrictions and applicability of each method

Innovation

• Bridge equation methods solve integral equations for causal targets
• Array decomposition recovers latent factors via eigenspace determination
• Contrasting model restrictions clarifies each method's applicability scope