Two-Step Mixed-Type Multivariate Bayesian Sparse Variable Selection with Shrinkage Priors

TL;DR

This paper presents a Bayesian method for multivariate regression with mixed outcomes, enabling variable selection and proving posterior contraction even when the number of covariates grows exponentially with sample size.

Contribution

It introduces a novel two-step approach for variable selection in high-dimensional mixed-type multivariate models with theoretical guarantees.

Findings

Method achieves sure screening property.

Proven posterior contraction under exponential p growth.

Validated through simulations and real data application.

Abstract

We introduce a Bayesian framework for mixed-type multivariate regression using continuous shrinkage priors. Our framework enables joint analysis of mixed continuous and discrete outcomes and facilitates variable selection from the $p$ covariates. Theoretical studies of Bayesian mixed-type multivariate response models have not been conducted previously and require more intricate arguments than the corresponding theory for univariate response models due to the correlations between the responses. In this paper, we investigate necessary and sufficient conditions for posterior contraction of our method when $p$ grows faster than sample size $n$. The existing literature on Bayesian high-dimensional asymptotics has focused only on cases where $p$ grows subexponentially with $n$. In contrast, we study the asymptotic regime where $p$ is allowed to grow exponentially in terms of $n$. We develop a novel two-step approach for variable selection which possesses the sure screening property and provably achieves posterior contraction even under exponential growth of $p$. We demonstrate the utility of our method through simulation studies and applications to real data, including a cancer genomics dataset where $n=174$ and $p=9183$. The R code to implement our method is available at https://github.com/raybai07/MtMBSP.