On default priors and approximate location models

A prior for statistical inference can be one of three basic types: a mathematical prior originally proposed in Bayes [Philos. Trans. R. Soc. Lond. 53 (1763) 370–418; 54 (1764) 269–325], a subjective prior presenting an opinion, or a truly objective prior based on an identified frequency reference. In this note we consider a method for deriving a mathematical prior based on approximate location models. This produces a mathematical posterior, and any practical interpretation of such a posterior is in terms of exact or approximate confidence under the postulated model. We describe how a proposed prior can be simply checked for consistency with confidence methods, using expansions about the maximum likelihood estimator.