A nonparametric empirical Bayes approach to large-scale multivariate regression

Multivariate regression has many applications, ranging from time series prediction to genomics. Borrowing information across the outcomes can improve prediction error, even when outcomes are statistically independent. Many methods exist to implement this strategy, for example the multiresponse lasso, but choosing the optimal method for a given dataset is difficult. These issues are addressed by establishing a connection between multivariate linear regression and compound decision problems. A nonparametric empirical Bayes procedure that can learn the optimal regression method from the data itself is proposed. Furthermore, the proposed procedure is free of tuning parameters and performs well in simulations and in a multiple stock price prediction problem.