On Finitely Additive Priors, Coherence, and Extended Admissibility

Abstract

A decision maker is seen to be coherent in the sense of de Finetti if, and only if, his probabilities are computed in accordance with some finitely additive prior. If a bounded loss function is specified, then a decision rule is extended admissible (i.e. and not uniformly dominated) if and only if it is {B}ayes for some finitely additive prior. However, if an improper countably additive prior is used, then decisions need not cohere and decision rules need not be extended admissible. Invariant, finitely additive priors are found and their posteriors calculated for a class of problems including translation parameter problems.