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Position: A Potential Outcomes Perspective on Pearl's Causal Hierarchy

Peng Wu, Linbo Wang

TL;DR

This paper reframes Pearl's causal hierarchy through a potential outcomes lens, providing a formal mapping of estimands to the second and third layers and a systematic discussion of identifiability under varying assumptions. It offers a distribution-based perspective that links structural causal models and potential outcomes, classifies a broad range of estimands with emphasis on cross-world and individual-level counterfactuals, and reviews identifiability strategies including data fusion, copula models, sensitivity analysis, and Pearl's three-step counterfactual inference. The work highlights the stronger assumptions required for third-layer estimands, discusses ranking and conformal inference methods for individual counterfactuals, and extends the framework to post-treatment settings such as principal causal effects, mediation, and counterfactual fairness. Collectively, the results guide researchers in selecting estimands aligned with scientific questions, assessing identifiability sufficiency, and applying appropriate strategies to address complex causal questions in practice.

Abstract

Pearl's causal hierarchy has garnered sustained attention as a foundational lens for formulating and understanding causal questions, and has been extensively discussed within the framework of structural causal models. In this paper, we revisit the hierarchy from a potential outcomes perspective and provide a formal, systematic classification of how various causal estimands are mapped to specific layers. Building on this classification, we summarize key identifiability challenges for estimands at different layers and review general strategies for achieving identification under varying assumptions. Our perspective is both intuitive and theoretically grounded, as higher layers of the hierarchy correspond to progressively richer features of the potential outcomes distribution, which in turn require stronger assumptions for identification. We expect this perspective to help clarify and deepen understanding of various causal estimands, particularly those in the third layer of the causal hierarchy, along with their associated identifiability challenges, identifiability strategies, and application scenarios.

Position: A Potential Outcomes Perspective on Pearl's Causal Hierarchy

TL;DR

This paper reframes Pearl's causal hierarchy through a potential outcomes lens, providing a formal mapping of estimands to the second and third layers and a systematic discussion of identifiability under varying assumptions. It offers a distribution-based perspective that links structural causal models and potential outcomes, classifies a broad range of estimands with emphasis on cross-world and individual-level counterfactuals, and reviews identifiability strategies including data fusion, copula models, sensitivity analysis, and Pearl's three-step counterfactual inference. The work highlights the stronger assumptions required for third-layer estimands, discusses ranking and conformal inference methods for individual counterfactuals, and extends the framework to post-treatment settings such as principal causal effects, mediation, and counterfactual fairness. Collectively, the results guide researchers in selecting estimands aligned with scientific questions, assessing identifiability sufficiency, and applying appropriate strategies to address complex causal questions in practice.

Abstract

Pearl's causal hierarchy has garnered sustained attention as a foundational lens for formulating and understanding causal questions, and has been extensively discussed within the framework of structural causal models. In this paper, we revisit the hierarchy from a potential outcomes perspective and provide a formal, systematic classification of how various causal estimands are mapped to specific layers. Building on this classification, we summarize key identifiability challenges for estimands at different layers and review general strategies for achieving identification under varying assumptions. Our perspective is both intuitive and theoretically grounded, as higher layers of the hierarchy correspond to progressively richer features of the potential outcomes distribution, which in turn require stronger assumptions for identification. We expect this perspective to help clarify and deepen understanding of various causal estimands, particularly those in the third layer of the causal hierarchy, along with their associated identifiability challenges, identifiability strategies, and application scenarios.
Paper Structure (15 sections, 16 equations, 2 figures, 3 tables)

This paper contains 15 sections, 16 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Pearl's Causal Hierarchy and Structural Causal Model
  5. A Potential Outcomes Perspective
  6. Overview
  7. Second Layer: Intervention
  8. Third Layer: Cross-World Causal Queries
  9. Causal Estimands
  10. Identifiability Challenges and Corresponding Strategies
  11. Third Layer: Individual-Level Counterfactual Queries
  12. Well-Definedness and Identifiability Challenges
  13. Identifiability and Estimation Strategies
  14. In the Presence of Post-Treatment Variable
  15. Conclusion

Figures (2)

  • Figure 1: Observed data for binary treatment, where $\checkmark$ and $?$ mean observed and unobserved, respectively. We omit $X$ for simplicity.
  • Figure 2: The proposed potential outcomes perspective on Pearl's causal hierarchy.

Theorems & Definitions (11)

  • Example 1: Binary Treatment, Second Layer
  • Example 2: Continuous Treatment, Second Layer
  • Example 3: Probability of Causation
  • Example 4: Treatment Benefit and Harm Rates
  • Example 5: Effect of Persuasion
  • Example 6: Distribution of ITE
  • Example 7: Risks of Decision-Making Based on CATE
  • Example 8: Principal Causal Effects
  • Example 9: Short-term and Long-term Treatment Effects
  • Example 10: Causal Fairness Metrics
  • ...and 1 more