Discovery and inference of possibly bi-directional causal relationships with invalid instrumental variables

Learning causal relationships between pairs of complex traits from observational studies is of great interest across various scientific domains. However, most existing methods assume the absence of unmeasured confounding and restrict causal relationships between two traits to be uni-directional, which may be violated in real-world systems. In this paper, we address the challenge of causal discovery and effect inference for two traits while accounting for unmeasured confounding and potential feedback loops. By leveraging possibly invalid instrumental variables, we provide identification conditions for causal parameters in a model that allows for bi-directional relationships, and we also establish identifiability of the causal direction under the introduced conditions. Then we propose a data-driven procedure to detect the causal direction and provide inference results about causal effects along the identified direction. We show that our method consistently recovers the true direction and produces valid confidence intervals for the causal effect. We conduct extensive simulation studies to show that our proposal outperforms existing methods. We finally apply our method to analyze real data sets from UK Biobank.