General Probabilities of Causation with Causal Knowledge

This paper addresses the question of whether additional causal knowledge can further tighten the bounds of probabilities of causation (PoCs) in multivalued settings, where treatments and outcomes can take multiple values. The authors extend existing bounds on PoCs that incorporate causal graph information from binary to multivalued settings, deriving new bounds for four representative forms of multivalued PoCs: PNS(k), PSub(k, p), PRep(k, q), and PN(k, p, q). These four forms are sufficient to represent any discrete PoC.

The paper considers multiple causal structures involving covariates that are not descendants of the treatment, covariates satisfying the back-door criterion, partial mediators, and pure mediators. For each graphical structure, the authors derive improved versions of all four existing bounds for multivalued probabilities of causation.

For non-descendant covariates, the paper presents Theorem 2, which provides new lower and upper bounds for PNS(k) by conditioning on the covariate Z and taking a weighted sum of stratum-specific bounds. Theorems 3, 4, and 5 similarly provide bounds for PSub(k, p), PRep(k, q), and PN(k, p, q) respectively. When Z also satisfies the back-door criterion, the back-door adjustment formula P (yj xj z) = P (yj xj, z) can be substituted into these theorems, requiring only observational data without the need for experimental data.

For mediation structures, the paper distinguishes between partial mediators (where X has a direct effect on Y) and pure mediators (where X affects Y only through Z). For partial mediators, Theorems 6-9 provide new upper bounds for PNS(k), PSub(k, p), PRep(k, q), and PN(k, p, q) respectively, while the lower bounds remain unchanged from those of Shu, Wang, and Li (2026). For pure mediators, Theorems 10-13 similarly provide new upper bounds for the four PoCs.

The paper illustrates the theoretical results with toy examples based on a pharmaceutical company developing a drug to prevent diabetes. In the first example, incorporating family history as a covariate (which satisfies the back-door criterion) tightens the bounds on PNS(3) from 0 ≤ PNS(3) ≤ 0.551 to 0 ≤ PNS(3) ≤ 0.033. In the second example, incorporating blood glucose status as a pure mediator tightens the bounds from 0 ≤ PNS(3) ≤ 0.507 to 0 ≤ PNS(3) ≤ 0.151.

Simulation studies demonstrate that the proposed bounds are tighter than existing nonbinary bounds. For the non-descendant covariate structure, the average improvement in the upper bound ranges from 0.0532 (for n=k=3) to 0.0557 (for n=k=5), with the percentage of samples benefiting from tighter bounds ranging from 89.08% to 99.85%. For the pure mediator structure, improvements range from 0.0583 (for n=k=3) to 0.0003 (for n=k=5), with 83.66% to 1.26% of samples benefiting. The partial mediator structure yields only small improvements, with the percentage of bounds tightened decreasing as dimension increases (from 8.2% for n=2, k=2 to 0% for n=4, k=4).

The paper concludes that incorporating causal graph information consistently yields tighter bounds than approaches based solely on experimental and observational distributions, with the non-descendant covariate structure providing the most substantial tightening as the dimension increases.

Improvements to AI Systems:

1. Causal Inference with Multivalued Treatments/Outcomes: Enhance AI systems to compute tighter bounds for probabilities of causation (e.g., PNS, PSub, PRep, PN) when treatments and outcomes are categorical with more than two levels, moving beyond binary-only causal reasoning.

2. Graph-Structure-Aware Bound Tightening: Enable AI to automatically leverage causal graph information (e.g., non-descendant covariates, back-door variables, mediators) to reduce uncertainty in causal effect estimates, producing narrower confidence intervals for decision-making.

3. Data-Efficient Causal Estimation: Improve AI systems to use observational data alone (via back-door adjustment) when experimental data is unavailable, while still achieving tighter bounds than naive observational methods—critical for domains like healthcare and economics where RCTs are costly.

4. Mediation-Aware Counterfactual Reasoning: Allow AI to distinguish between partial and pure mediators, refining upper bounds on causal probabilities when direct and indirect effects coexist, leading to more accurate what-if predictions.

5. Scalable Causal Bounds for High-Dimensional Settings: Equip AI with algorithms that maintain bound-tightening performance as the number of treatment/outcome categories grows (e.g., n=k=5), avoiding the degradation seen in naive approaches.

6. Automated Causal Structure Discovery for Bounds: Integrate the paper's theorems into AI pipelines that automatically identify which covariates are non-descendants or mediators from data, then apply the corresponding bound formulas without manual specification.

7. Robust Decision Support Under Partial Identifiability: Improve AI systems to output ranges of causal probabilities (instead of point estimates) when full identifiability is impossible, using the derived bounds to quantify epistemic uncertainty—useful for risk assessment and policy planning.

8. Simulation-Guided Bound Selection: Enable AI to run simulations (as in the paper) to pre-select the causal structure (e.g., non-descendant vs. mediator) that yields the greatest bound tightening for a given dataset, optimizing inference efficiency.

9. Domain-Specific Applications: Apply the improved bounds to AI systems in pharmaceutical development (e.g., drug efficacy with multiple dosage levels), public health (multi-level risk factors), and personalized medicine (multi-valued treatment responses) to make more precise causal claims from observational data.

10. Explainable AI for Causal Claims: Enhance AI to output not only tighter bounds but also the causal graph structure used, making the reasoning transparent and auditable—crucial for regulatory compliance in medicine and finance.

Probabilities of causation (PoCs) characterize individual causal responses that cannot be directly observed and therefore generally require partial identification. Tian and Pearl first derived theoretically sharp bounds for binary PoCs, including the probability of necessity (PN), the probability of sufficiency (PS), and the probability of necessity and sufficiency (PNS). Mueller et al. subsequently tightened the bounds for binary PNS by incorporating causal information encoded in covariates and mediators. More recently, Li and Pearl, as well as Shu et al., extended PoCs to multivalued settings and derived corresponding theoretical bounds. These developments naturally raise the question of whether additional causal knowledge can further tighten the bounds in multivalued settings. This paper addresses this question by deriving tighter bounds for multivalued PoCs through the incorporation of causal information encoded in covariates and mediators. We illustrate the theoretical results with toy examples, while simulation studies further demonstrate that the proposed bounds are tighter than existing nonbinary bounds.