Linear SCM Identification in the Presence of Confounders and Gaussian Noise

Noisy linear structural causal models (SCMs) in the presence of confounding variables are known to be identifiable if all confounding and noise variables are non-Gaussian and unidentifiable if all are Gaussian. The identifiability when only some are Gaussian remains concealed. We show that, in the presence of Gaussian noise, a linear SCM is uniquely identifiable provided that \emph{(i)} the number of confounders is at most the number of the observed variables, \emph{(ii)} the confounders do not have a Gaussian component, and \emph{(iii)} the causal structure of the SCM is known. If the third condition is relaxed, the SCM becomes finitely identifiable; more specifically, it belongs to a set of at most $n!$ linear SCMS, where $n$ is the number of observed variables. The confounders in all of these $n!$ SCMs share the same joint probability distribution function (PDF), which we obtain analytically. For the case where both the noise and confounders are Gaussian, we provide further insight into the existing counter-example-based unidentifiability result and demonstrate that every SCM with confounders can be represented as an SCM without confounders but with the same joint PDF.