Learning distribution-free anchored linear structural equation models in the presence of measurement error

Abstract This study tackles the challenge of identifiability in distribution-free anchored linear structural equation models (SEMs), where the observed variables are imperfect measures for the target variables, and the error distributions are not restricted to being Gaussian. It introduces the geometry-faithfulness assumption, ensuring that partial correlations serve as direct indicators of d-separation/connection. The study establishes the identifiability of distribution-free anchored linear SEMs under the same identifiability conditions for anchored Gaussian linear SEMs, but by replacing the faithfulness assumption with the geometry-faithfulness assumption. Moreover, it shows that the learning algorithm leveraging the PC algorithm with Fisher’s z-test, originally designed for anchored Gaussian linear SEMs, remains applicable and effective for distribution-free anchored linear SEMs. It also provides statistical guarantees for the proposed algorithm, including the strong geometry-faithfulness assumption, ensuring its consistency. These theoretical contributions are validated through extensive numerical experiments and the analysis of real galaxy data.