Reduced-rank regression recognises the possibility of a rank-deficient matrix of coefficients, which is particularly useful when the data is high-dimensional. We propose a novel Bayesian model for estimating the rank of the rank of the coefficient matrix, which obviates the need of 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, 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：低秩回归考虑了系数矩阵可能秩不足的情况，这在处理高维数据时尤其实用。我们提出了一种新颖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，并允许进行不确定性量化。我们的方法对回归系数矩阵采用混合先验，并对其低秩分解施加全局-局部收缩先验。然后，我们利用信号自适应变量选择器执行稀疏化处理，并定义了两种新工具：后验包含概率不确定性指数和相关指数。通过模拟研究验证了该方法的有效性，并在烟草化学成分与星系测光的实际数据应用中展示了其优势与实用性。