Precision Requirements for Monte Carlo Sums within Hierarchical Bayesian Inference

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.