A class of proper scoring functions which combine the error in a decision problem and the precision of the statistical decision rule is introduced. The Bayesian procedures with respect to these loss functions are pairs formed by the usual Bayes decision and by the expected posterior loss. A necessary and sufficient condition for admissibility under the corresponding risk is given.
"Loss Functions for Loss Estimation." Ann. Statist. 16 (3) 1262 - 1269, September, 1988. https://doi.org/10.1214/aos/1176350960