Admissibility of a posterior predictive decision rule

As reviewed by [Owhadi and Scovel], the field of statistical decision theory introduced by Wald, building on a game theoretic foundation developed by von Neumann and Morgenstern, provides links between Bayesian and frequentist statistical philosophies through the concepts of decision rules, admissibility, and risk functions amongst others. Moreover, a recent thrust of research motivated by machine learning has put much emphasis on prediction problems for which Bayesian methodology has been widely used. The purpose of this note is to demonstrate that classic decision theoretic results can be simply applied to the analysis of prediction problems. In fact, both [Berger] and [Robert] remark upon the ease of applying statistical decision theory within the context of prediction, however, no explicit result is stated in either work; the contribution of this note, therefore, is to highlight a simple way in which the results of statistical decision theory might apply to prediction problems. To the author's knowledge the most similar lines of thought appear in work by [Nayak and El-Baz], where the loss function depends on the underlying parameter (in contrast to what follows).