We introduce a new empirical Bayes approach for large-scale multiple linear regression. Our approach combines two key ideas: (i) the use of flexible "adaptive shrinkage" priors, which approximate the nonparametric family of scale mixture of normal distributions by a finite mixture of normal distributions; and (ii) the use of variational approximations to efficiently estimate prior hyperparameters and compute approximate posteriors. Combining these two ideas results in fast and flexible methods, with computational speed comparable to fast penalized regression methods such as the Lasso, and with superior prediction accuracy across a wide range of scenarios. Furthermore, we show that the posterior mean from our method can be interpreted as solving a penalized regression problem, with the precise form of the penalty function being learned from the data by directly solving an optimization problem (rather than being tuned by cross-validation). Our methods are implemented in an R package, mr.ash.alpha, available from https://github.com/stephenslab/mr.ash.alpha

翻译：我们提出了一种适用于大规模多元线性回归的新经验贝叶斯方法。该方法融合了两个关键思想：（i）使用灵活的“自适应收缩”先验，该先验通过正态分布的有限混合来近似正态分布尺度混合的非参数族；（ii）利用变分逼近高效估计先验超参数并计算近似后验。结合这两个思想，我们得到了快速且灵活的方法，其计算速度可与Lasso等快速惩罚回归方法相媲美，并且在广泛场景下具有更优的预测精度。此外，我们证明该方法的后验均值可解释为求解一个惩罚回归问题，其中惩罚函数的具体形式通过直接求解优化问题从数据中学习（而非通过交叉验证调优）。我们的方法已在R包mr.ash.alpha中实现，可从https://github.com/stephenslab/mr.ash.alpha获取。