PAC-Causal Learnability: From Identification to Finite-Sample Guarantees

arXiv:2507.18903v2 Announce Type: replace-cross Abstract: 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 $\epsilon$ with probability at least $1-\delta$. 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.