The problem of estimating many means from noisy data is a fundamental challenge with applications across disciplines. In these so-called "compound decision problems" it is often possible to do better than estimating each mean separately, by "borrowing strength" from all of the data to estimate each mean. One way to construct better estimators is Empirical Bayes (EB), which posits that the means come from a common (but unknown) prior and attempts to approximate the Bayes estimator using all of the data. In this talk, we show how black box regressors can be used to construct EB estimators when replicates are available for each mean. Our approach, called AURORA, can approximately achieve the Bayes risk, without any assumptions on the prior or the likelihood. We also show how data fission (Leiner et al., 2023) can be used to extend our approach to the case where replicates are not available–that is, when there is only one observation for each mean.
This is joint work with Nikos Ignatiadis, Sujayam Saha, and Omkar Muralidharan.