While the classical credibility theory addresses the important problem of combining claim experience and prior information to update the prediction for loss, it does not provide a very satisfactory solution. The method is based on arbitrary selection of the coverage probability and the accuracy parameter. Furthermore, for tractability some restrictive assumptions about the loss distribution have to be imposed.

Bühlmann credibility theory sets the problem in a rigorous statistical framework of optimal prediction, using the least mean squared error criterion. It is flexible enough to incorporate various distributional assumptions of loss variables. The approach is further extended to enable the claim experience of different blocks of policies with different exposures to be combined for improved forecast through the Bühlmann–Straub model.

The Bühlmann and Bühlmann–Straub models recognize the interaction of two sources of variability in the data, namely the variation due to between-group differences and variation due to within-group fluctuations. We begin this chapter with the set-up of the Bühlmann credibility model, and a review of how the variance of the loss variable is decomposed into between-group and within-group variations. We derive the Bühlmann credibility factor and updating formula as the minimum mean squared error predictor. The approach is then extended to the Bühlmann–Straub model, in which the loss random variables have different exposures.