A hierarchical Bayesian approach that permits simultaneous inference for the regression coefficient matrix and the error precision (inverse covariance) matrix in the multivariate linear model is proposed. Assuming a natural ordering of the elements of the response, the precision matrix is reparameterized so it can be estimated with univariate-response linear regression techniques. A novel generalized bridge regression prior that accommodates both sparse and dense settings and is competitive with alternative methods for univariate-response regression is proposed and used in this framework. Two component-wise Markov chain Monte Carlo algorithms are developed for sampling, including a data augmentation algorithm based on a scale mixture of normals representation. Numerical examples demonstrate that the proposed method is competitive with comparable joint mean-covariance models, particularly in estimation of the precision matrix. The method is also used to estimate the 253 by 253 precision matrix of 90,670 spectra extracted from images taken by the Hubble Space Telescope, demonstrating its computational feasibility for problems with large n and q.

翻译：本文提出了一种分层贝叶斯方法，用于在多元线性模型中同时对回归系数矩阵和误差精度（逆协方差）矩阵进行推断。假设响应变量元素存在自然顺序，通过对精度矩阵进行重参数化，使其能够使用单变量响应线性回归技术进行估计。在此框架中，提出并采用了一种新颖的广义桥回归先验，该先验既能适应稀疏设置也能适应密集设置，并且在单变量响应回归中与现有方法相比具有竞争力。开发了两种分量式马尔可夫链蒙特卡洛算法进行采样，包括一种基于正态分布尺度混合表示的数据增强算法。数值算例表明，所提方法与同类联合均值-协方差模型相比具有竞争力，特别是在精度矩阵估计方面。该方法还被用于估计从哈勃太空望远镜拍摄图像中提取的90,670个光谱所对应的253×253精度矩阵，证明了其在大n和大q问题中的计算可行性。