  Variable selection has played a critical role in modern statistical learning
and scientific discoveries. Numerous regularization and Bayesian variable
selection methods have been developed in the past two decades for variable
selection, but most of these methods consider selecting variables for only one
response. As more data is being collected nowadays, it is common to analyze
multiple related responses from the same study. Existing multivariate variable
selection methods select variables for all responses without considering the
possible heterogeneity across different responses, i.e. some features may only
predict a subset of responses but not the rest. Motivated by the multi-trait
fine mapping problem in genetics to identify the causal variants for multiple
related traits, we developed a novel multivariate Bayesian variable selection
method to select critical predictors from a large number of grouped predictors
that target at multiple correlated and possibly heterogeneous responses. Our
new method is featured by its selection at multiple levels, its incorporation
of prior biological knowledge to guide selection and identification of best
subset of responses predictors target at. We showed the advantage of our method
via extensive simulations and a real fine mapping example to identify causal
variants associated with different subsets of addictive behaviors.
