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

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.

💡 Research Summary

The paper tackles a long‑standing limitation in causal inference: existing analytical results for probabilities of causation (PoCs) have been confined to binary treatments and outcomes, while many real‑world applications involve multi‑valued (categorical or ordinal) variables. Building on the structural causal model (SCM) framework, the authors derive explicit closed‑form upper and lower bounds for a representative family of discrete PoCs that cover the general case where the treatment X takes m possible values and the outcome Y takes n possible values.

The key methodological innovations are threefold. First, the authors introduce equivalence classes of PoCs, showing that any arbitrary discrete PoC can be reduced to a small set of canonical forms without loss of expressive power. This reduction relies on a systematic mapping that pads or truncates value sets while preserving the underlying causal relationships. Second, they establish a replaceability principle: within a given PoC expression, any individual treatment‑outcome pair can be swapped for another pair, and the same bound formulas still apply. This property enables rapid transformation of complex queries into the canonical forms for which bounds are already known.

The canonical family consists of four types of PoCs:

1. PNS(k) – the probability that k distinct treatment‑outcome pairs all occur simultaneously (a higher‑order generalisation of the classic probability of necessity and sufficiency).
2. PSub(k, p) – the joint probability that k treatment‑outcome pairs occur together with an additional treatment assignment p.
3. PRep(k, q) – the joint probability that k treatment‑outcome pairs occur together with an additional outcome q.
4. PN(k, p, q) – the joint probability of k treatment‑outcome pairs together with a specific treatment p and outcome q.

For each of these, Theorems 2‑5 present bounds that involve only simple arithmetic operations (max, min, sums) over observable quantities: the experimental causal effects (P(y_i|x_j)) and the observational joint distribution (P(x_j, y_i)). The bounds are sound for any dimensionality; the authors provide a formal proof that the lower bound never exceeds the true PoC and the upper bound never falls below it.

To assess tightness, the authors apply Balke’s linear‑programming method—known to yield optimal bounds for binary cases—to low‑dimensional settings (3‑valued and 4‑valued treatments/outcomes). In all tested instances, the closed‑form bounds coincide exactly with the LP optimum, establishing empirical tightness in these cases. While a general proof of tightness for arbitrary dimensions remains an open conjecture, the empirical evidence strongly suggests that the bounds are optimal or near‑optimal even as the number of values grows.

The paper also provides a thorough computational comparison with the recent recursive bounds of Li & Pearl (2024). Across a range of simulated SCMs, the new bounds are consistently narrower (typically 5–10 % tighter) and require orders of magnitude less computation time because they avoid recursive evaluation and large linear programs.

A concrete application is presented in the medical domain: three antihypertensive treatments (x₁, x₂, x₃) and three outcome categories (poor, moderate, good). The authors estimate the PoC (P(y_3x_1, y_1x_2, y_2x_3)), which quantifies the proportion of patients who would experience a good outcome under treatment 1, a poor outcome under treatment 2, and a moderate outcome under treatment 3—all simultaneously in counterfactual worlds. Using experimental trial data (900 patients, evenly split across treatments) and observational self‑selection data, the closed‑form bounds are computed (e.g., lower bound ≈ 0.509, upper bound ≈ 0.732). This illustrates how multi‑valued PoCs can reveal nuanced, patient‑specific treatment trade‑offs that are invisible to marginal outcome rates.

Limitations are acknowledged. The tightness proof is currently limited to low‑dimensional cases; extending it to arbitrary m and n remains an open theoretical challenge. The bounds assume accurate estimates of experimental and observational distributions, which may be problematic with small samples or heavy measurement noise. Moreover, the analysis is restricted to discrete variables; extending the framework to continuous treatments or outcomes would require additional technical development.

In summary, the paper delivers the first non‑recursive, closed‑form analytical bounds for multi‑valued probabilities of causation, supported by rigorous soundness proofs, empirical tightness verification, and demonstrable practical relevance. By reducing any discrete PoC to a small canonical set and providing a simple replaceability mechanism, the work dramatically simplifies the computation of counterfactual quantities needed for personalized decision‑making in healthcare, finance, energy, and beyond. Future work will likely focus on proving tightness in full generality, handling continuous variables, and integrating Bayesian uncertainty quantification to further enhance applicability.