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 where the number of covariates $p$ may be larger than sample size $n$. 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 $n$. We further establish that subexponential growth is both a necessary and a sufficient condition 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. Numerical experiments and analyses of real datasets demonstrate the ability of our joint modeling approach to improve predictive accuracy and identify significant variables in multivariate mixed response models. 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类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型结果，并在协变量个数$p$可能大于样本量$n$时实现变量选择。该模型可通过吉布斯采样算法实现，所有条件分布均具有可处理形式，从而得到简单的单步估计流程。在$p$随$n$呈次指数增长时，我们推导了单步估计量的后验收缩率。进一步证明，次指数增长是单步估计量实现后验一致性的充要条件。随后引入适用于大$p$情况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$p$随$n$呈指数增长时，我们的两步估计量能够确凿实现后验收缩，从而突破单步估计量的局限。数值实验与真实数据分析表明，我们的联合建模方法能够提升多元混合响应模型的预测精度，并识别出显著变量。方法对应的R代码参见https://github.com/raybai07/MtMBSP。