Before revision:  We consider posterior estimators of such effects , where the average is computed conditional on the observation sample. While in various settings it is common to "shrink" individual estimates -- e.g., of teacher value-added or hospital quality -- toward a common mean to reduce estimation noise, a study of the frequentist properties of posterior average estimators is lacking.
After revision:  We consider posterior estimators of such effects , 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.