Bayesian Bidirectional Mendelian Randomization Under Correlated and Uncorrelated Pleiotropy Using GWAS Summary Statistics.

Mendelian randomization (MR) uses genetic variants as instruments to study causal relationships between traits. Standard unidirectional MR approaches are vulnerable to pleiotropy and typically assume that pleiotropic effects are independent of instrument strength, assumptions that may fail when causation operates in both directions. We propose BayBiMR, a Bayesian bidirectional MR framework that jointly estimates causal effects in both directions while modeling both correlated and uncorrelated pleiotropy through a hierarchical spike-and-slab prior on pleiotropic effects and an Inverse-Wishart prior on the joint covariance of direct and pleiotropic SNP effects. The likelihood is based on a small-feedback reduced-form approximation to the underlying simultaneous-equation model, which preserves conjugacy and enables efficient closed-form posterior computation. Posterior inference is carried out via a blocked Gibbs sampler, and a data-perturbation extension, BayBiMR(DP), is provided to improve empirical frequentist calibration in finite samples. Simulation studies show that BayBiMR and BayBiMR(DP) provide better type I error control than existing methods in the settings considered, particularly under correlated and mixed pleiotropy. Applications to large-scale GWAS summary statistics on white-matter (WM) microstructures, BMI, sleep duration, and neuroticism illustrate the method's practical utility as an exploratory and sensitivity tool.