On Bernoulli experiments with imprecise prior probabilities

This paper deals with a situation where the prior distribution in a Bayesian treatment of a Bernoulli experiment is not known precisely. Imprecision for Bernoulli experiments is discussed for the case where the prior densities are defined in the form of intervals of measures, and a simple model with conjugate imprecise prior densities is used for ease of calculation in updating. Attention is focused on imprecision with regard to predictive probabilities. The main aim of the paper is an analysis of imprecision, emphasizing the important relationship between information and imprecision. On the basis of this relationship, a rule for updating the set of prior distributions is proposed that is different from the theory advocated by Walley and suggests a new area for research. Imprecision within the chosen model is compared with intuition, and, although a complete lack of prior information cannot be represented perfectly, it can be approximated well.