Stepwise Bayes distributional inference for a finite population

Stepwise Bayes arguments can be used to derive various decision rules. Admissibility of such rules follows if additional conditions are satisfied. We show that in complete generality almost admissibility is in place. A uniform distribution example is used to demonstrate how stepwise Bayes arguments can be used when the support of the observation distribution depends on the unknown parameter. We then discuss distributional inference and show that weighted Polya posterior distributions provide admissible distributional inference for finite population problems when strictly proper loss functions are used.