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Identification of Probabilities of Causation: from Recursive to Closed-Form Bounds

Xin Shu, Shuai Wang, Ang Li

TL;DR

This work extends probabilities of causation to multi-valued treatments and outcomes within Structural Causal Models by deriving closed-form, non-recursive bounds for a representative family of discrete PoCs. It introduces equivalence classes and a replaceability principle to unify and transfer bounds across value permutations, proving soundness in all dimensions and verifying tightness in low dimensions via Balke’s linear programming. Empirically, the proposed bounds consistently tighten the recursive Li–Pearl bounds while remaining computationally simpler, enabling efficient, heterogeneity-aware counterfactual attribution. The results are illustrated with toy medical and educational examples, highlighting practical impact in cross-world counterfactual reasoning and personalized decision-making. The work also outlines open questions, including a general completeness proof for all dimensions and potential extensions with covariates and causal graphs.

Abstract

Probabilities of causation (PoCs) are fundamental quantities for counterfactual analysis and personalized decision making. However, existing analytical results are largely confined to binary settings. This paper extends PoCs to multi-valued treatments and outcomes by deriving closed form bounds for a representative family of discrete PoCs within Structural Causal Models, using standard experimental and observational distributions. We introduce the notion of equivalence classes of PoCs, which reduces arbitrary discrete PoCs to this family, and establish a replaceability principle that transfers bounds across value permutations. For the resulting bounds, we prove soundness in all dimensions and empirically verify tightness in low dimensional cases via Balke's linear programming method; we further conjecture that this tightness extends to all dimensions. Simulations indicate that our closed form bounds consistently tighten recent recursive bounds while remaining simpler to compute. Finally, we illustrate the practical relevance of our results through toy examples.

Identification of Probabilities of Causation: from Recursive to Closed-Form Bounds

TL;DR

This work extends probabilities of causation to multi-valued treatments and outcomes within Structural Causal Models by deriving closed-form, non-recursive bounds for a representative family of discrete PoCs. It introduces equivalence classes and a replaceability principle to unify and transfer bounds across value permutations, proving soundness in all dimensions and verifying tightness in low dimensions via Balke’s linear programming. Empirically, the proposed bounds consistently tighten the recursive Li–Pearl bounds while remaining computationally simpler, enabling efficient, heterogeneity-aware counterfactual attribution. The results are illustrated with toy medical and educational examples, highlighting practical impact in cross-world counterfactual reasoning and personalized decision-making. The work also outlines open questions, including a general completeness proof for all dimensions and potential extensions with covariates and causal graphs.

Abstract

Probabilities of causation (PoCs) are fundamental quantities for counterfactual analysis and personalized decision making. However, existing analytical results are largely confined to binary settings. This paper extends PoCs to multi-valued treatments and outcomes by deriving closed form bounds for a representative family of discrete PoCs within Structural Causal Models, using standard experimental and observational distributions. We introduce the notion of equivalence classes of PoCs, which reduces arbitrary discrete PoCs to this family, and establish a replaceability principle that transfers bounds across value permutations. For the resulting bounds, we prove soundness in all dimensions and empirically verify tightness in low dimensional cases via Balke's linear programming method; we further conjecture that this tightness extends to all dimensions. Simulations indicate that our closed form bounds consistently tighten recent recursive bounds while remaining simpler to compute. Finally, we illustrate the practical relevance of our results through toy examples.

Paper Structure

This paper contains 27 sections, 6 theorems, 99 equations, 3 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Key Contributions
  3. Preliminaries
  4. Main Results
  5. Probability of necessity and sufficiency($k$)
  6. Probability of substitute($k,p$)
  7. Probability of replacement($k,q$)
  8. Probability of necessity($k,p,q$)
  9. Equivalence Class in Probability of Causation
  10. Replaceability in Probability of Causation
  11. Examples
  12. Cross-Treatment Heterogeneity in Medicine
  13. Personalized Decision-Making in Educational Interventions
  14. Simulated Results
  15. Dimension sweep ($n\in[3,20]$).
  16. ...and 12 more sections

Key Result

Theorem 2

Suppose 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 necessity and sufficiency($k$) $P({y_{1}}_{x_{1}},...,{y_{k}}_{x_{k}})$ is bounded as following:

Figures (3)

  • Figure 1: Bounds differences between Li--Pearl and Theorem 2.
  • Figure 2: $\mathrm{PNS}$ bounds over 100 instances for $n=k=3$.
  • Figure 3: Per-instance $\mathrm{PNS}$ bounds across different $(n,k)$ settings.

Theorems & Definitions (13)

  • Definition 1: Probabilities of Causation
  • Theorem 2: Probability of necessity and sufficiency($k$) (PNS($k$))
  • Theorem 3: Probability of substitute($k,p$) (PSub($k,p$))
  • Theorem 4: Probability of replacement($k,q$) (PRep($k,q$))
  • Theorem 5: Probability of necessity($k,p,q$) (PN($k,p,q$))
  • Theorem 6: Equivalence classes in probabilities of causation
  • Theorem 7: Replaceability in probabilities of causation
  • proof
  • proof
  • proof
  • ...and 3 more