  Large-scale randomized experiments, sometimes called A/B tests, are
increasingly prevalent in many industries. Though such experiments are often
analyzed via frequentist $t$-tests, arguably such analyses are deficient:
$p$-values are hard to interpret and not easily incorporated into
decision-making. As an alternative, we propose an empirical Bayes approach,
which assumes that the treatment effects are realized from a "true prior". This
requires inferring the prior from previous experiments. Following Robbins, we
estimate a family of marginal densities of empirical effects, indexed by the
noise scale. We show that this family is characterized by the heat equation. We
develop a spectral maximum likelihood estimate based on a Fourier series
representation, which can be efficiently computed via convex optimization. In
order to select hyperparameters and compare models, we describe two model
selection criteria. We demonstrate our method on simulated and real data, and
compare posterior inference to that under a Gaussian mixture model of the
prior.
