Bayesian Multivariable Bidirectional Mendelian Randomization

Mendelian randomization (MR) is a pivotal tool in genetic epidemiology, leveraging genetic variants as instrumental variables to infer causal relationships between modifiable exposures and health outcomes. Traditional MR methods, while powerful, often rest on stringent assumptions such as the absence of feedback loops, which are frequently violated in complex biological systems. In addition, many popular MR approaches focus on only two variables (i.e., one exposure and one outcome) whereas our motivating applications have many variables. In this article, we introduce a novel Bayesian framework for \emph{multivariable} MR that concurrently addresses \emph{unmeasured confounding} and \emph{feedback loops}. Central to our approach is a sparse conditional cyclic graphical model with a sparse error variance-covariance matrix. Two structural priors are employed to enable the modeling and inference of causal relationships as well as latent confounding structures. Our method is designed to operate effectively with summary-level data, facilitating its application in contexts where individual-level data are inaccessible, e.g., due to privacy concerns. It can also account for horizontal pleiotropy. Through extensive simulations and applications to the GTEx and OneK1K data, we demonstrate the superior performance of our approach in recovering biologically plausible causal relationships in the presence of possible feedback loops and unmeasured confounding. The R package that implements the proposed method is available at \texttt{MR.RGM}.