Bounding Probabilities of Causation with Partial Causal Diagrams

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.

• The paper develops an optimization framework that encodes non-descendant covariates, back-door sets, and mediators as constraints on joint counterfactual distributions to bound PNS, PN, and PS without a complete causal diagram.
• The method extends prior bounds to arbitrary finite-valued treatments and outcomes and can combine covariate-specific data even when joint covariate distributions are unavailable.
• Simulations show narrower bounds than Tian–Pearl methods in up to 996 of 1,000 cases, while mediator-based programs improve average bound gaps but introduce nonconvex computational challenges.

Motivation and problem setting

Probabilities of causation (PoCs)—the probability of necessity (PN), probability of sufficiency (PS), and probability of necessity and sufficiency (PNS)—quantify individual-level counterfactual claims such as whether a treatment was necessary or sufficient for an observed outcome. Under the structural causal model (SCM) semantics, these quantities involve joint events over multiple potential outcomes (e.g., $P(y_x, y'_{x'})$) that cannot be jointly observed, so they are not point-identifiable in general. The classical response is to derive bounds: Tian and Pearl obtained sharp bounds from experimental and observational distributions alone [tian2000probabilities], with a later theoretical proof by Li and Pearl [li2019unit], while subsequent work showed that a fully specified causal diagram can narrow them further [pearl:etal21-r505, zhang2022partial].

The paper under review identifies a practical gap between these two extremes. Graph-based tightening results typically assume complete knowledge of the causal structure and are largely restricted to binary treatments and outcomes; the no-assumption Tian–Pearl bounds, while valid, are often too wide for decision making. Real-world domain knowledge is frequently partial—e.g., knowledge that certain variables are non-descendants of the treatment, that a set satisfies the back-door criterion, or that a mediator exists—without a fully specified graph. The paper proposes an optimization-based framework in which such partial causal information is encoded directly as constraints on the joint counterfactual distribution, yielding valid bounds without full identifiability.

Framework

The approach follows Balke's linear programming methodology [balke1995probabilistic]: decision variables $p_{j_1 \ldots j_K}$ represent probabilities over the joint counterfactual distribution $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$, and available information is imposed as linear constraints. Bounds on PNS($k$) — defined as $P(y_{1\,x_1}, \ldots, y_{k\,x_k})$, following the multivalued characterization of Shu, Wang, and Li [shu2025identification] — are then obtained by minimizing and maximizing an appropriate objective over the feasible polytope.

Three main results are established:

• Theorem 1 (non-descendant covariates, multivalued). Given $m$ covariates $Z_1,\ldots,Z_m$ containing no descendant of $X$ in a partial causal diagram $G$, constraints encode both experimental quantities $P({y_s}_{x_t}, z_{ir_i} \text{ for } Z_i \in Z)$ and observational quantities $p_{j_1 \ldots j_K}$0 for arbitrary subsets $p_{j_1 \ldots j_K}$1 of jointly available covariates. The special case $p_{j_1 \ldots j_K}$2 recovers Theorem 4 of Mueller, Li, and Pearl [pearl:etal21-r505]. Crucially, treatments and outcomes may take arbitrary finite values, removing the binary restriction of prior work.
• Corollary 1 (covariate-specific data only). When joint distributions over covariates are unavailable and only per-covariate data $p_{j_1 \ldots j_K}$3 and $p_{j_1 \ldots j_K}$4 exist, the framework still integrates information across all covariates simultaneously rather than treating each separately. This modularity is a substantive departure from prior work, which required access to $p_{j_1 \ldots j_K}$5 and $p_{j_1 \ldots j_K}$6 for every value of a single covariate set.
• Theorem 2 (back-door set plus mediator). With $p_{j_1 \ldots j_K}$7 satisfying the back-door criterion and $p_{j_1 \ldots j_K}$8 a mediator of $p_{j_1 \ldots j_K}$9 and $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$0, the program augments the counterfactual variables with $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$1 and adds bilinear equality constraints enforcing the conditional independencies $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$2 and $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$3, which follow from the twin-network representation. This unifies confounder and mediator information in a single program; Mueller–Li–Pearl's binary illustration with an empty back-door set (their Theorem 6) is a special case.

A notable scope remark: although theorems are stated for PNS, other PoCs require only changing the objective function—the constraint set is unchanged, so PN and PS bounds follow at no additional structural cost. Multiple mediators can be handled either by extending the theorem to vectors or via the equivalent-mediator construction of Li et al. [li2022bounds].

Empirical evaluation

Simulations compare against the Tian–Pearl (TP) bounds and the Mueller–Li–Pearl (MLP) bounds on 1,000 randomly generated distributions compatible with each causal structure, restricted to binary settings because both baselines are binary-only.

Multiple non-descendant covariates. With only covariate-specific data ($P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$4 covariates, $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$5), Corollary 1 improves upon TP bounds for 634 of 1,000 samples at $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$6, rising to 996 at $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$7 (>90% for $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$8); improvements over MLP reach 818 of 1,000 at $P(Y_{x_1}, \ldots, Y_{x_n}, \ldots)$9. Average bound-gap reduction relative to TP is roughly 0.02–0.04 on both sides. The average TP gap grows from 0.2136 ($k$0) to 0.3965 ($k$1), whereas the proposed gap grows only from 0.1622 to 0.3165. Relative to MLP, gains are modest but monotonically increasing in $k$2 (average lower-bound improvement rising from 0 at $k$3 to 0.0043 at $k$4).

Back-door set with mediator. For the confounder-plus-mediator structure, Theorem 2 achieves an average gap of 0.2308 versus 0.2833 (TP) and 0.2417 (MLP), improving TP bounds for 782/1,000 samples and MLP bounds for 245/1,000. The qualitative point here is that MLP cannot incorporate confounder and mediator information simultaneously, whereas the proposed formulation does so through additional constraints.

These results support the paper's central claim: aggregating heterogeneous partial causal information systematically tightens PoC bounds even when no joint covariate distributions and no complete causal diagram are available.

Limitations and open questions

The paper is candid about computational limits. Theorems 1 and Corollary 1 yield linear programs solvable efficiently for moderate discrete domains, but Theorem 2's conditional-independence constraints are bilinear equalities, making the optimization nonconvex—with increased runtime and sensitivity to local optima as state spaces grow. Whether tractable relaxations, alternative parameterizations, or aggregated mediator summaries can recover most of the tightening effect with certified bounds remains open. Two further caveats bear on interpretation: the empirical comparisons are confined to binary settings despite the framework's multivalued generality, and the mediator experiments cover only a single mediator, with multi-mediator effects left unexamined. Additionally, the validity of the bounds rests on the correctness of the supplied partial assumptions (e.g., that $k$5 truly contains no descendants of $k$6 or forms a valid back-door set); the paper does not address robustness to misspecified partial knowledge.

Conclusion

This paper formulates bounding of probabilities of causation as constrained optimization over counterfactual distributions, into which partial causal knowledge—non-descendant covariates, back-door sets, mediators—is encoded modularly as constraints. It strictly generalizes the Tian–Pearl and Mueller–Li–Pearl bounds to arbitrary finite-valued treatments and outcomes, permits use of covariate-specific data without joint distributions, and combines confounder and mediator information within one program. Simulations show consistent gap reductions over both baselines, with improvements covering the large majority of sampled distributions. The main unresolved issue is computational scalability of the nonlinear mediator-augmented programs, which the authors identify as the key direction for future work.