Causal Effect Identifiability in the Presence of Latent Confounders Without Auxiliary Variables

It is a fundamental challenge to ascertain whether the causal effect of a treatment on an outcome is identifiable in the presence of latent confounders, which serves as the logical prerequisite for recovering the causal effect in a partially observed system. While prior literature demonstrates that the causal effect is identifiable when there exist auxiliary variables subject to stringent structural constraints, this paper investigates identifiability of the causal effect without such variables. This means that we ground identifiability solely in the joint distribution of the treatment-outcome pair, which constitutes the irreducible statistical basis for causal effect identification. Focusing on linear structural causal models (SCMs), we provide a nuanced and complete characterization of identifiability of the causal effect contingent on the distributional properties of exogenous noises. Specifically, we formulate a set of mutually exclusive and collectively exhaustive conditions regarding the Gaussianity of exogenous noises, ascertain under which conditions the causal effect is identifiable and under which it is not, while also quantifying the cardinality of the feasible solution set for the unidentifiable cases. Finally, we empirically validate our theoretical findings.