Abstract

The normal means problem is very simple: given normally-distributed observations with known variances and unknown means, estimate the means. That is, given X_j \sim N(\theta_j, \sigma_j^2, estimate \theta_j. A key idea is that one can do better than the maximum likelihood estimates, \hat{\theta}_j= \X_j, in particular by use of appropriate “shrinkage” estimators. One attractive way to perform shrinkage estimation in practice is to use Empirical Bayes methods. That is, to assume that \theta_j are independent and identically distributed from some distribution g that is to be estimated from the data. Then, given such an estimate \hat{g}, the posterior distributions of \theta_j can be computed to perform inference. We call this the “Empirical Bayes Normal Means” (EBNM) problem.

Despite its simplicity, solving the EBNM problem has a wide range of practical applications. Here we present some flexible non-parametric approaches we have recently developed for solving the EBNM problem, and describe their application to several different settings: false discovery rate (FDR) estimation, non-parametric smoothing, and sparse matrix factorization problems (ie sparse factor analysis and sparse principal components analysis).