Exact credibility reference Bayesian premiums

Abstract

In this paper, reference analysis, the tool provided by Berger et al. (2009), is used to obtain reference Bayesian premiums, which can be helpful when the practitioner has insufficient information to elicit a prior distribution. The Bayesian premiums thus obtained are based exclusively on prior distributions built from the model generated and from the available data. This mechanism produces an objective Bayesian inference, which appears to be the same as the robust Γ-minimax inference. In an informational-theoretical sense, the prior distribution used to make the inference is less informative. These Bayesian premiums are expected to approximate those which would have been obtained using proper priors describing a vague initial state of knowledge. Useful credibility expressions are readily derived by taking classes of priors involving restrictions on moments, i.e., restrictions on the collective or prior premium when the weighted squared-error loss function is used.