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We introduce a Bayesian framework for mixed-type multivariate regressionusing continuous shrinkage priors. Our framework enables joint analysis ofmixed continuous and discrete outcomes and facilitates variable selection fromthe $p$ covariates. Theoretical studies of Bayesian mixed-type multivariateresponse models have not been conducted previously and require more intricatearguments than the corresponding theory for univariate response models due tothe correlations between the responses. In this paper, we investigate necessaryand sufficient conditions for posterior contraction of our method when $p$grows faster than sample size $n$. The existing literature on Bayesianhigh-dimensional asymptotics has focused only on cases where $p$ growssubexponentially with $n$. In contrast, we consider the asymptotic regime where$p$ is allowed to grow exponentially in terms of $n$. We develop a noveltwo-step approach for variable selection which possesses the sure screeningproperty and provably achieves posterior contraction even under exponentialgrowth of $p$. We demonstrate the utility of our method through simulationstudies and applications to real datasets. The R code to implement our methodis available at https://github.com/raybai07/MtMBSP.

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