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 consider 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 datasets. The R code to implement our method is available at https://github.com/raybai07/MtMBSP.

翻译：我们提出了一种基于连续收缩先验的混合型多元回归贝叶斯框架。该框架能够联合分析混合连续与离散响应变量，并促进从 $p$ 个协变量中进行变量选择。先前尚未有关于混合型多元响应模型的贝叶斯理论研究，且由于响应变量间的相关性，此类模型需要比单变量响应模型更复杂的论证。本文研究了当 $p$ 超过样本量 $n$ 时，该方法后验收缩性的必要与充分条件。现有关于贝叶斯高维渐近性的文献仅关注 $p$ 随 $n$ 呈次指数增长的情形。相比之下，我们考虑了 $p$ 允许随 $n$ 呈指数增长的新近渐近框架。我们开发了一种新颖的两步变量选择方法，该方法具有确定筛选性质，并能在 $p$ 呈指数增长的情况下证明性地实现后验收缩。通过模拟研究和实际数据集应用，我们展示了该方法的实用性。实现本方法的R代码可在 https://github.com/raybai07/MtMBSP 获取。