Many modern statistical problems can be cast in the framework of multivariate regression, where the main task is to make statistical inference for a possibly sparse and low-rank coefficient matrix. The low-rank structure in the coefficient matrix is of intrinsic multivariate nature, which, when combined with sparsity, can further lift dimension reduction, conduct variable selection, and facilitate model interpretation. Using a Bayesian approach, we develop a unified sparse and low-rank multivariate regression method to both estimate the coefficient matrix and obtain its credible region for making inference. The newly developed sparse and low-rank prior for the coefficient matrix enables rank reduction, predictor selection and response selection simultaneously. We utilize the marginal likelihood to determine the regularization hyperparameter, so our method maximizes its posterior probability given the data. For theoretical aspect, the posterior consistency is established to discuss an asymptotic behavior of the proposed method. The efficacy of the proposed approach is demonstrated via simulation studies and a real application on yeast cell cycle data.

中文翻译：

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贝叶斯稀疏降秩多元回归

许多现代统计问题可以在多元回归的框架内进行，其中主要任务是对可能稀疏和低秩系数矩阵进行统计推断。系数矩阵中的低秩结构具有内在的多元性质，结合稀疏性可以进一步提升降维，进行变量选择，便于模型解释。使用贝叶斯方法，我们开发了一种统一的稀疏和低秩多元回归方法来估计系数矩阵并获得其用于推理的可信区域。新开发的系数矩阵的稀疏和低秩先验可以同时进行降阶、预测变量选择和响应选择。我们利用边际似然来确定正则化超参数，所以我们的方法在给定数据的情况下最大化其后验概率。在理论方面，建立后验一致性来讨论所提出方法的渐近行为。通过模拟研究和酵母细胞周期数据的实际应用证明了所提出方法的有效性。