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Bounding Probabilities of Causation with Partial Causal Diagrams

Yuxuan Xie, Ang Li

TL;DR

This work addresses individualized causal questions by bounding probabilities of causation (PoCs) under partial causal information. It introduces an optimization-based framework that encodes structural and auxiliary information as constraints on counterfactual distributions, generalizing prior bounds to multi-valued treatments/outcomes and modularly utilizing covariates and mediator data. The proposed bounds consistently tighten compared to Tian–Pearl and Mueller–Li–Pearl baselines, with larger gains when more partial information is available, as demonstrated in simulations with non-descendant covariates and back-door mediators. By enabling meaningful counterfactual reasoning without full identifiability, the approach enhances applicability of PoCs in medicine, policy, and law, and suggests directions for scalable, interpretable bounds under incomplete causal knowledge.

Abstract

Probabilities of causation are fundamental to individual-level explanation and decision making, yet they are inherently counterfactual and not point-identifiable from data in general. Existing bounds either disregard available covariates, require complete causal graphs, or rely on restrictive binary settings, limiting their practical use. In real-world applications, causal information is often partial but nontrivial. This paper proposes a general framework for bounding probabilities of causation using partial causal information. We show how the available structural or statistical information can be systematically incorporated as constraints in a optimization programming formulation, yielding tighter and formally valid bounds without full identifiability. This approach extends the applicability of probabilities of causation to realistic settings where causal knowledge is incomplete but informative.

Bounding Probabilities of Causation with Partial Causal Diagrams

TL;DR

This work addresses individualized causal questions by bounding probabilities of causation (PoCs) under partial causal information. It introduces an optimization-based framework that encodes structural and auxiliary information as constraints on counterfactual distributions, generalizing prior bounds to multi-valued treatments/outcomes and modularly utilizing covariates and mediator data. The proposed bounds consistently tighten compared to Tian–Pearl and Mueller–Li–Pearl baselines, with larger gains when more partial information is available, as demonstrated in simulations with non-descendant covariates and back-door mediators. By enabling meaningful counterfactual reasoning without full identifiability, the approach enhances applicability of PoCs in medicine, policy, and law, and suggests directions for scalable, interpretable bounds under incomplete causal knowledge.

Abstract

Probabilities of causation are fundamental to individual-level explanation and decision making, yet they are inherently counterfactual and not point-identifiable from data in general. Existing bounds either disregard available covariates, require complete causal graphs, or rely on restrictive binary settings, limiting their practical use. In real-world applications, causal information is often partial but nontrivial. This paper proposes a general framework for bounding probabilities of causation using partial causal information. We show how the available structural or statistical information can be systematically incorporated as constraints in a optimization programming formulation, yielding tighter and formally valid bounds without full identifiability. This approach extends the applicability of probabilities of causation to realistic settings where causal knowledge is incomplete but informative.
Paper Structure (15 sections, 3 theorems, 27 equations, 9 figures, 2 tables)

This paper contains 15 sections, 3 theorems, 27 equations, 9 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Preliminaries and Related Work
  4. Bounds with Partial Causal Information
  5. Non-descendant Covariates
  6. Backdoor Set with Mediators
  7. Simulation Results
  8. Multiple non-descendant covariates with covariate-specific data
  9. Backdoor Set with Mediators
  10. Conclusion
  11. Limitations and future work.
  12. Appendix
  13. Proof of Theorem \ref{['tm1']}
  14. Proof of Theorem \ref{['tm2']}
  15. Proof of Theorem \ref{['tm3']}

Key Result

Theorem 3.1

Given a partial causal diagram $G$ and distribution compatible with $G$, let $Z_1,...,Z_m$ be a set of $m$ variables that does not contain any descendant of $X$ in $G$. Let the variable $X$ has $n$ values $x_1,...,x_n$ and $Y$ has $n$ values $y_1,...,y_n$, $k \le n$, then the probability of necessit where LB and UB is the min and max solution to the following linear optimization problem, where the

Figures (9)

  • Figure 1: Causal structures used in the experiments.
  • Figure 2: PNS lower bounds for $n=6$ in Figure \ref{['causalg1']}.
  • Figure 3: PNS lower bounds for $n=6$ in Figure \ref{['causalg1']}.
  • Figure 4: PNS lower bounds for $n=5$ in Figure \ref{['causalg1']}.
  • Figure 5: PNS lower bounds for $n=5$ in Figure \ref{['causalg1']}.
  • ...and 4 more figures

Theorems & Definitions (9)

  • Definition 2.1: Probability of necessity (PN)
  • Definition 2.2: Probability of sufficiency (PS)
  • Definition 2.3: Probability of necessity and sufficiency (PNS)
  • Theorem 3.1
  • Corollary 3.2
  • Theorem 3.3
  • proof
  • proof
  • proof