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Empirical Bayes is attractive in such settings, but standard parametric approaches rest on assumptions about the form of the prior distribution which can be hard to justify and which introduce unnecessary tuning parameters. We extend the nonparametric maximum-likelihood estimator (NPMLE) for Gaussian location mixture densities to allow for multivariate, heteroscedastic errors. 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. 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 denoizing without prior knowledge. We provide finite-sample bounds on the average Hellinger accuracy of an NPMLE for estimating the marginal densities of the observations. We also demonstrate the adaptive and nearly optimal properties of NPMLEs for deconvolution. We apply our method to two denoizing problems in astronomy and to two hierarchical linear modelling problems in social science and biology.<\/jats:p>","DOI":"10.1093\/jrsssb\/qkae040","type":"journal-article","created":{"date-parts":[[2024,5,29]],"date-time":"2024-05-29T18:15:46Z","timestamp":1717006546000},"page":"1-32","source":"Crossref","is-referenced-by-count":10,"title":["Multivariate, heteroscedastic empirical Bayes via nonparametric maximum likelihood"],"prefix":"10.1093","volume":"87","author":[{"given":"Jake A","family":"Soloff","sequence":"first","affiliation":[{"name":"Department of Statistics, University of Chicago , Chicago, IL ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Adityanand","family":"Guntuboyina","sequence":"additional","affiliation":[{"name":"Department of Statistics, University of California at Berkeley, Berkeley , CA ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bodhisattva","family":"Sen","sequence":"additional","affiliation":[{"name":"Department of Statistics, Columbia University , Room 1032, 1255 Amsterdam Avenue, New York, NY 10027 ,","place":["USA"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"286","published-online":{"date-parts":[[2024,5,29]]},"reference":[{"issue":"2","key":"2025021015291522400_qkae040-B1","doi-asserted-by":"publisher","first-page":"42","DOI":"10.3847\/1538-4365\/aa9e8a","article-title":"The fourteenth data release of the sloan digital sky survey: First Spectroscopic data from the extended baryon oscillation spectroscopic survey and from the second phase of the Apache point observatory galactic evolution experiment","volume":"235","author":"Abolfathi","year":"2018","journal-title":"The Astrophysical Journal Supplement Series"},{"key":"2025021015291522400_qkae040-B2","author":"Aibar","year":"2023"},{"issue":"2","key":"2025021015291522400_qkae040-B3","doi-asserted-by":"publisher","first-page":"706","DOI":"10.1086\/177901","article-title":"Linear regression for astronomical data with measurement errors and intrinsic scatter","volume":"470","author":"Akritas","year":"1996","journal-title":"The Astrophysical Journal"},{"issue":"3","key":"2025021015291522400_qkae040-B4","doi-asserted-by":"publisher","first-page":"587","DOI":"10.1093\/imaiai\/iaz013","article-title":"Maximum number of modes of Gaussian mixtures","volume":"9","author":"Am\u00e9ndola","year":"2020","journal-title":"Information and Inference: A Journal of the IMA"},{"issue":"4","key":"2025021015291522400_qkae040-B5","doi-asserted-by":"publisher","first-page":"145","DOI":"10.3847\/1538-3881\/aad7bf","article-title":"Improving Gaia parallax precision with a data-driven model of stars","volume":"156","author":"Anderson","year":"2018","journal-title":"The Astronomical Journal"},{"key":"2025021015291522400_qkae040-B6","doi-asserted-by":"publisher","author":"Banerjee","year":"2023","DOI":"10.48550\/arXiv.2002.12586"},{"issue":"1","key":"2025021015291522400_qkae040-B7","doi-asserted-by":"publisher","first-page":"57","DOI":"10.1016\/0378-3758(85)90024-2","article-title":"Numerical estimation of a probability measure","volume":"11","author":"B\u00f6hning","year":"1985","journal-title":"Journal of Statistical Planning and Inference"},{"key":"2025021015291522400_qkae040-B8","volume-title":"Computer-assisted analysis of mixtures and applications: Meta-analysis, disease mapping and others","author":"B\u00f6hning","year":"1999"},{"issue":"3","key":"2025021015291522400_qkae040-B9","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1023\/A:1024222817645","article-title":"The EM algorithm with gradient function update for discrete mixtures with known (fixed) number of components","volume":"13","author":"B\u00f6hning","year":"2003","journal-title":"Statistics and Computing"},{"issue":"2B","key":"2025021015291522400_qkae040-B10","doi-asserted-by":"publisher","first-page":"1657","DOI":"10.1214\/10-AOAS439","article-title":"Extreme deconvolution: Inferring complete distribution functions from noisy, heterogeneous and incomplete observations","volume":"5","author":"Bovy","year":"2011","journal-title":"The Annals of Applied Statistics"},{"key":"2025021015291522400_qkae040-B11","doi-asserted-by":"publisher","first-page":"A2","DOI":"10.1051\/0004-6361\/201629512","article-title":"Gaia data release 1. 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