  We revisit empirical Bayes in the absence of a tractable likelihood function,
as is typical in scientific domains relying on computer simulations. We
investigate how the empirical Bayesian can make use of neural density
estimators first to use all noise-corrupted observations to estimate a prior or
source distribution over uncorrupted samples, and then to perform
single-observation posterior inference using the fitted source distribution. We
propose an approach based on the direct maximization of the log-marginal
likelihood of the observations, examining both biased and de-biased estimators,
and comparing to variational approaches. We find that, up to symmetries, a
neural empirical Bayes approach recovers ground truth source distributions.
With the learned source distribution in hand, we show the applicability to
likelihood-free inference and examine the quality of the resulting posterior
estimates. Finally, we demonstrate the applicability of Neural Empirical Bayes
on an inverse problem from collider physics.
