  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.
