Bayes double sample procedures are given for estimating the mean of exponential family distributions. The distribution can be multiparameter and so the case of normal mean with unknown variance is included. Linear combination loss functions are usually assumed. Stein's double sample procedure, and other intuitive double sample procedures for estimating a binomial parameter, are studied. Recommendations are given, including guidelines for the size of the initial samples.
"Bayes Double Sample Estimation Procedures." Ann. Statist. 12 (3) 1035 - 1049, September, 1984. https://doi.org/10.1214/aos/1176346719