Reduced-rank regression recognises the possibility of a rank-deficient matrix of coefficients. We propose a novel Bayesian model for estimating the rank of the coefficient matrix, which obviates the need for post-processing steps and allows for uncertainty quantification. Our method employs a mixture prior on the regression coefficient matrix along with a global-local shrinkage prior on its low-rank decomposition. Then, we rely on the Signal Adaptive Variable Selector to perform sparsification and define two novel tools: the Posterior Inclusion Probability uncertainty index and the Relevance Index. The validity of the method is assessed in a simulation study, and then its advantages and usefulness are shown in real-data applications on the chemical composition of tobacco and on the photometry of galaxies.

翻译：低秩回归允许回归系数矩阵可能是低秩的。我们提出了一种新的贝叶斯模型来估计系数矩阵的秩，该模型无需后处理步骤，并能进行不确定性量化。我们的方法采用混合先验作用于回归系数矩阵，并结合全局-局部收缩先验对其低秩分解进行建模。随后，我们利用信号自适应变量选择器进行稀疏化，并定义了两种新工具：后验包含概率不确定性指数和相关性指数。通过模拟研究评估了该方法的有效性，并在烟草化学成分和星系光度测量的实际数据应用中展示了其优势和实用性。