Multivariate, Heteroscedastic Empirical Bayes via Nonparametric Maximum Likelihood

• Activity Number: 360 - New Areas in Complex High-Dimensional Data Analysis
• Type: Invited
• Date/Time: Wednesday, August 10, 2022 : 8:30 AM to 10:20 AM
• Sponsor: International Indian Statistical Association
• Author(s): Bodhisattva Sen* and Adityanand Guntuboyina and Jake Soloff
• Companies: Columbia University and University of California Berkeley and University of California at Berkeley
• Keywords: Deconvolution; Denoising; Posterior means

Abstract

Multivariate, heteroscedastic errors complicate statistical inference in many large-scale denoising problems. We study Empirical Bayes methodology in this setting by considering the nonparametric maximum likelihood estimator (NPMLE) for Gaussian location mixture densities in this problem. NPMLEs estimate an arbitrary prior by solving an infinite-dimensional, convex optimization problem; we show that this convex optimization problem can be tractably approximated by a finite-dimensional version. We study the characterization and uniqueness of the NPMLE in the multivariate setting. The empirical Bayes posterior means based on an NPMLE have low regret, meaning they closely target the oracle posterior means one would compute with the true prior in hand. We prove an oracle inequality implying that the empirical Bayes estimator performs at nearly the optimal level (up to logarithmic factors) for denoising without prior knowledge. We also demonstrate the adaptive and nearly-optimal properties of NPMLEs for deconvolution. We apply the method to an astronomy dataset, constructing a fully data-driven color-magnitude diagram of 1.4 million stars in the Milky Way.