The Credible Distribution

Credibility theory is concerned with the problem of forecasting the mean performance (claim frequency, total losses, etc.) of an individual risk, selected from a collective of heterogeneous risks, based upon the statistics of the collective, and upon the individual's experience data. The classic results, derived by American actuaries in the 1920's, were further strengthened by Bailey and Mayerson in 1950 and 1965, who showed that these results were exact Bayesian for certain risk distributions. Bühlmann, in 1967, then showed that the credibility formulae were the best least-squares linearized approximation to the exact Bayesian forecast, for general risk distributions. This paper extends credibility theory to the problem of forecasting the distribution of individual risk, based upon collective statistics and individual experience data. Although the problem is, in principle, solved by finding a Bayesian conditional distribution, this approach requires a detailed knowledge of collective structure. The credible distribution, on the other hand, requires fewer prior statistics, and is also a best least-squares linearized approximation to the exact Bayesian distribution. Following the theoretical development, detailed computational results are given.