  Hierarchical Bayesian inference is often conducted with estimates of the
target distribution derived from Monte Carlo sums over samples from separate
analyses of parts of the hierarchy or from mock observations used to estimate
sensitivity to a target population. We investigate requirements on the number
of Monte Carlo samples needed to guarantee the estimator of the target
distribution is precise enough that it does not affect the inference. We
consider probabilistic models of how Monte Carlo samples are generated, showing
that the finite number of samples introduces additional uncertainty as they act
as an imperfect encoding of the components of the hierarchical likelihood.
Additionally, we investigate the behavior of estimators marginalized over
approximate measures of the uncertainty, comparing their performance to the
Monte Carlo point estimate. We find that correlations between the estimators at
nearby points in parameter space are crucial to the precision of the estimate.
Approximate marginalization that neglects these correlations will either
introduce a bias within the inference or be more expensive (require more Monte
Carlo samples) than an inference constructed with point estimates. We therefore
recommend that hierarchical inferences with empirically estimated target
distributions use point estimates.
