Seemingly unrelated regression is a natural framework for regressing multiple correlated responses on multiple predictors. The model is very flexible, with multiple linear regression and covariance selection models being special cases. However, its practical deployment in genomic data analysis under a Bayesian framework is limited due to both statistical and computational challenges. The statistical challenge is that one needs to infer both the mean vector and the inverse covariance matrix, a problem inherently more complex than separately estimating each. The computational challenge is due to the dimensionality of the parameter space that routinely exceeds the sample size. We propose the use of horseshoe priors on both the mean vector and the inverse covariance matrix. This prior has demonstrated excellent performance when estimating a mean vector or inverse covariance matrix separately. The current work shows these advantages are also present when addressing both simultaneously. A full Bayesian treatment is proposed, with a sampling algorithm that is linear in the number of predictors. MATLAB code implementing the algorithm is freely available from github at https://github.com/liyf1988/HS_GHS. Extensive performance comparisons are provided with both frequentist and Bayesian alternatives, and both estimation and prediction performances are verified on a genomic data set.

似乎不相关的回归是用于回归多个预测变量上多个相关响应的自然框架。该模型非常灵活，特殊情况有多个线性回归和协方差选择模型。但是，由于统计和计算方面的挑战，在贝叶斯框架下进行基因组数据分析的实际应用受到了限制。统计上的挑战是，既需要推论均值矢量又要推销协方差逆矩阵，这是一个固有的问题，比分别估计每个问题要复杂得多。计算上的挑战是由于参数空间的维数通常超过样本大小。我们建议在均值向量和逆协方差矩阵上都使用马蹄先验。当分别估计均值向量或逆协方差矩阵时，该先验数据已显示出出色的性能。当前的工作表明，同时解决这两个问题时也具有这些优势。提出了一种完整的贝叶斯处理方法，其采样算法的预测变量数量呈线性。可以从github上免费获得实现该算法的MATLAB代码，网址为https://github.com/liyf1988/HS_GHS。同时提供了频繁性和贝叶斯选择的广泛性能比较，并且在基因组数据集上验证了估计性能和预测性能。可从github上免费获得实现该算法的MATLAB代码，网址为https://github.com/liyf1988/HS_GHS。同时提供了频繁性和贝叶斯选择的广泛性能比较，并且在基因组数据集上验证了估计性能和预测性能。可从github上免费获得实现该算法的MATLAB代码，网址为https://github.com/liyf1988/HS_GHS。同时提供了频繁性和贝叶斯选择的广泛性能比较，并且在基因组数据集上验证了估计性能和预测性能。