A model of prior ignorance for inferences in the one-parameter exponential family

Abstract

This paper proposes a model of prior ignorance about a scalar variable based on a set of distributions M. In particular, a set of minimal properties that a set M of distributions should satisfy to be a model of prior ignorance without producing vacuous inferences is defined. In the case the likelihood model corresponds to a one-parameter exponential family of distributions, it is shown that the above minimal properties are equivalent to a special choice of the domains for the parameters of the conjugate exponential prior. This makes it possible to define the largest (that is, the least-committal) set of conjugate priors M that satisfies the above properties. The obtained set M is a model of prior ignorance with respect to the functions (queries) that are commonly used for statistical inferences; it is easy to elicit and, because of conjugacy, tractable; it encompasses frequentist and the so-called objective Bayesian inferences with improper priors. An application of the model to a problem of inference with count data is presented.