On the Granularity of Causal Effect Identifiability

The classical notion of causal effect identifiability is defined in terms of treatment and outcome variables. In this note, we consider the identifiability of state-based causal effects: how an intervention on a particular state of treatment variables affects a particular state of outcome variables. We demonstrate that state-based causal effects may be identifiable even when variable-based causal effects may not. Moreover, we show that this separation occurs only when additional knowledge -- such as context-specific independencies and conditional functional dependencies -- is available. We further examine knowledge that constrains the states of variables, and show that such knowledge does not improve identifiability on its own but can improve both variable-based and state-based identifiability when combined with other knowledge such as context-specific independencies. Our findings highlight situations where causal effects of interest may be estimable from observational data and this identifiability may be missed by existing variable-based frameworks.