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 paper, 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 -- is available. We further examine knowledge that constrains the states of variables, and show that such knowledge can improve both variable-based and state-based identifiability when combined with other knowledge such as context-specific independencies. We finally propose an approach for identifying causal effects under these additional constraints, and conduct empirical studies to further illustrate the separations between the two levels of identifiability.