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.

翻译：我们提出了一种使用连续收缩先验的混合型多元回归贝叶斯框架。该框架能够对连续与离散结果进行联合分析，并有助于从$p$个协变量中进行变量选择。由于响应变量之间的相关性，贝叶斯混合型多元响应模型的理论研究此前尚未开展，且比单变量响应模型的相应理论需要更复杂的论证。本文研究了当$p$的增长速度快于样本量$n$时，我们方法的后验收缩的充分必要条件。现有关于贝叶斯高维渐近理论的文献仅关注$p$相对于$n$呈次指数增长的情况。相比之下，我们研究了允许$p$相对于$n$呈指数增长的渐近体系。我们开发了一种新颖的两步变量选择方法，该方法具备确定筛选性质，并能在$p$呈指数增长的情况下仍可证明实现后验收缩。我们通过模拟研究及真实数据应用（包括一个$n=174$、$p=9183$的癌症基因组学数据集）展示了我们方法的实用性。实现我们方法的R代码可在 https://github.com/raybai07/MtMBSP 获取。