# Bayesian Variable Selection for Multi-Outcome Models Through Shared Shrinkage

## TL;DR

This paper introduces a Bayesian variable selection method for multivariate regression models that leverages shared shrinkage priors to identify key covariates across multiple outcomes, improving selection accuracy.

## Contribution

It extends global-local shrinkage priors to multivariate models, enabling simultaneous covariate selection across multiple response variables with a novel prior structure.

## Key findings

- Effective in identifying important covariates across outcomes
- Performs well in simulation studies
- Demonstrated on real data example

## Abstract

Variable selection over a potentially large set of covariates in a linear model is quite popular. In the Bayesian context, common prior choices can lead to a posterior expectation of the regression coefficients that is a sparse (or nearly sparse) vector with a few non-zero components, those covariates that are most important. This article extends the global-local shrinkage idea to a scenario where one wishes to model multiple response variables simultaneously. Here, we have developed a variable selection method for a K-outcome model (multivariate regression) that identifies the most important covariates across all outcomes. The prior for all regression coefficients is a mean zero normal with coefficient-specific variance term that consists of a predictor-specific factor (shared local shrinkage parameter) and a model-specific factor (global shrinkage term) that differs in each model. The performance of our modeling approach is evaluated through simulation studies and a data example.