Empirical Bayes Estimation and Inference via Smooth Nonparametric Maximum Likelihood

The empirical Bayes $g$-modeling approach based on the nonparametric maximum likelihood estimator (NPMLE) has been central to large-scale estimation and inference in the normal means problem. However, theoretical guarantees for uncertainty quantification remain scarce. A key obstacle is that the NPMLE is necessarily discrete, which yields discrete posterior credible sets and a slow logarithmic deconvolution rate. We address both limitations by introducing a hierarchical Gaussian smoothing layer that restricts the mixing distribution to a Gaussian location mixture. Our smooth NPMLE inherits the favorable properties of the classical NPMLE: it is computable via convex optimization and achieves nearly parametric denoising performance. Moreover, it achieves a polynomial deconvolution rate that is asymptotically minimax over the corresponding class. Our procedure also leads to estimated smooth posteriors that converge to the true posteriors at a polynomial rate. Further, we characterize marginal coverage sets that are optimal in expected length, construct plug-in estimators of these sets, and establish theoretical guarantees for the estimated sets in terms of both coverage probability and expected length. We also extend the theory to settings with model misspecification and heteroscedastic Gaussian observations, and study identifiability of the proposed hierarchical model.