$\epsilon$-Identifiability of Causal Quantities

Identifying the effects of causes and causes of effects is vital in virtually every scientific field. Often, however, the needed probabilities may not be fully identifiable from the available data sources. This paper shows how approximate identifiability is still possible for several probabilities of causation. We term this $\epsilon\text{-identifiability}$ and demonstrate its usefulness in cases where the behavior of certain subpopulations can be restricted within sufficiently narrow bounds. In particular, we show how unidentifiable causal effects and counterfactual probabilities can be $\epsilon\text{-identified}$ when such allowances are made. Often, these allowances are easily measured and reasonably assumed. Finally, $\epsilon\text{-identifiability}$ is applied to the unit selection problem.