Generalized Independent Noise Condition for Estimating Latent Variable Causal Graphs

Causal discovery aims to recover causal structures or models underlying the observed data. Despite its success in certain domains, most existing methods focus on causal relations between observed variables, while in many scenarios the observed ones may not be the underlying causal variables (e.g., image pixels), but are generated by latent causal variables or confounders that are causally related. To this end, in this paper, we consider Linear, Non-Gaussian Latent variable Models (LiNGLaMs), in which latent confounders are also causally related, and propose a Generalized Independent Noise (GIN) condition to estimate such latent variable graphs. Specifically, for two observed random vectors \mathbf{Y} and \mathbf{Z}, GIN holds if and only if \omega {\intercal}\mathbf{Y} and \mathbf{Z} are statistically independent, where \omega is a parameter vector characterized from the cross-covariance between \mathbf{Y} and \mathbf{Z} . From the graphical view, roughly speaking, GIN implies that causally earlier latent common causes of variables in \mathbf{Y} d-separate \mathbf{Y} from \mathbf{Z} .