  The results from Genome-Wide Association Studies (GWAS) on thousands of
phenotypes provide an unprecedented opportunity to infer the causal effect of
one phenotype (exposure) on another (outcome). Mendelian randomization (MR), an
instrumental variable (IV) method, has been introduced for causal inference
using GWAS data. Due to the polygenic architecture of complex traits/diseases
and the ubiquity of pleiotropy, however, MR has many unique challenges compared
to conventional IV methods. We propose a Bayesian weighted Mendelian
randomization (BWMR) for causal inference to address these challenges. In our
BWMR model, the uncertainty of weak effects owing to polygenicity has been
taken into account and the violation of IV assumption due to pleiotropy has
been addressed through outlier detection by Bayesian weighting. To make the
causal inference based on BWMR computationally stable and efficient, we
developed a variational expectation-maximization (VEM) algorithm. Moreover, we
have also derived an exact closed-form formula to correct the posterior
covariance which is often underestimated in variational inference. Through
comprehensive simulation studies, we evaluated the performance of BWMR,
demonstrating the advantage of BWMR over its competitors. Then we applied BWMR
to make causal inference between 130 metabolites and 93 complex human traits,
uncovering novel causal relationship between exposure and outcome traits. The
BWMR software is available at https://github.com/jiazhao97/BWMR.
