Statistical Inferences Based on a Second-Order Possibility Distribution

Abstract A new, general method of statistical inference is proposed. It encompasses all the coherent forms of statistical inference that can be derived from a Bayesian prior distribution, Bayesian sensitivity analysis or upper and lower prior probabilities. The method is to model prior uncertainty about statistical parameters in terms of a second-order possibility distribution (a special type of upper probability) which measures the plausibility of each conceivable prior probability distribution. This defines an imprecise hierarchical model. Two,applications are studied: the problem of robustifying Bayesian analyses by forming a neighbourhood of a Bayesian prior distribution, and the problem of combining prior opinions from different sources.