Economists are often interested in estimating averages with respect to distributions of unobservables. Examples are moments of individual fixed-effects, average partial effects in discrete choice models, and counterfactual simulations in structural models. For such quantities, we propose and study posterior average effects (PAE), where the average is computed conditional on the sample, in the spirit of empirical Bayes and shrinkage methods. While the usefulness of shrinkage for prediction is well-understood, a justification of posterior conditioning to estimate population averages is currently lacking. We show that PAE have minimum worst-case bias under local misspecification of the parametric distribution of unobservables. This provides a rationale for reporting these estimators in applications. We introduce a measure of informativeness of the posterior conditioning, which quantifies the bias of PAE relative to parametric model-based estimators, and we study other robustness properties of PAE for estimation and prediction. As illustrations, we report PAE estimates of distributions of neighborhood effects in the US, and of permanent and transitory components in a model of income dynamics.