General sample size analysis for probabilities of causation: a delta method approach

Probabilities of causation (PoCs) have developed steadily, from the binary measures first introduced by Pearl to the non-binary setting recently studied by Shu et al. As the theory has matured, PoCs have found applications in a growing range of areas, including personalized medicine, marketing and legal attribution. Shu et al. proposed closed-form bounds for multi-valued treatments and outcomes and proved them valid, but left their sharpness in high dimensions open. We close this gap with a mathematical proof. We further show that $\PNS$, $\PSub$, $\PRep$ and $\PN$ are all special cases of a single, more general formula. Combined with their equivalence classes, this settles the question of sharp bounds for discrete PoCs from experimental and observational data: in every dimension, they are given in closed form by a single theorem.

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.

Causal identification asks whether a causal target can be recovered from the observational distribution in an idealized infinite-data setting. In practice, only finite samples are available, and identifiability alone does not determine how much data are needed for reliable estimation. We introduce Probably Approximately Correct Causal (PACC) learnability, which formalizes uniform accuracy-confidence guarantees over a causal model class. From a PAC-learning perspective, we ask how many samples are sufficient to estimate a causal target to accuracy $ε$ with probability at least $1-δ$. We show that PACC sample complexity is shaped by three recurring factors: causal identification strength, target complexity, and nuisance complexity, and derive a general sufficient bound linking them to finite-sample causal learning. We illustrate the framework through self-controlled case series, instrumental variables, and propensity-score-based estimation, showing how distinct causal methods can be analyzed within a common framework.