We introduce a Bayesian framework for mixed-type multivariate regression using shrinkage priors. Our method enables joint analysis of mixed continuous and discrete outcomes and facilitates variable selection from the $p$ covariates. Our model can be implemented with a Gibbs sampling algorithm where all conditional distributions are tractable, leading to a simple one-step estimation procedure. We derive the posterior contraction rate for the one-step estimator when $p$ grows subexponentially with respect to sample size $n$. We further establish that subexponential growth is both necessary and sufficient for the one-step estimator to achieve posterior consistency. We then introduce a two-step variable selection approach that is suitable for large $p$. We prove that our two-step algorithm possesses the sure screening property. Moreover, our two-step estimator can provably achieve posterior contraction even when $p$ grows exponentially in $n$, thus overcoming a limitation of the one-step estimator. We demonstrate the utility of our method through simulation studies and applications to real datasets. R codes to implement our method are available at https://github.com/raybai07/MtMBSP.

翻译：我们提出了一种基于收缩先验的混合型多元回归贝叶斯框架。该方法能够联合分析混合连续与离散结果，并从$p$个协变量中实现变量选择。模型可通过吉布斯采样算法实现，其中所有条件分布均为可解析形式，从而得到简单的单步估计过程。我们推导了当$p$相对于样本量$n$呈亚指数增长时，单步估计量的后验收缩率。进一步证明了亚指数增长是单步估计量实现后验一致性的充要条件。随后，我们提出适用于大规模$p$的两步变量选择方法，并证明该算法具有确定筛选性质。此外，当$p$关于$n$呈指数增长时，两步估计量仍可证明实现后验收缩，从而克服了单步估计量的局限。通过模拟研究与真实数据集应用，我们展示了该方法的实用性。实现本方法的R代码详见https://github.com/raybai07/MtMBSP。