In this chapter we consider the Bayesian approach in updating the prediction for future losses. We consider the derivation of the posterior distribution of the risk parameters based on the prior distribution of the risk parameters and the likelihood function of the data. The Bayesian estimate of the risk parameter under the squared-error loss function is the mean of the posterior distribution. Likewise, the Bayesian estimate of the mean of the random loss is the posterior mean of the loss conditional on the data.

In general, the Bayesian estimates are difficult to compute, as the posterior distribution may be quite complicated and intractable. There are, however, situations where the computation may be straightforward, as in the case of conjugate distributions. We define conjugate distributions and provide some examples for cases that are of relevance in analyzing loss measures. Under specific classes of conjugate distributions, the Bayesian predictor is the same as the Bühlmann predictor. Specifically, when the likelihood belongs to the linear exponential family and the prior distribution is the natural conjugate, the Bühlmann credibility estimate is equal to the Bayesian estimate. This result provides additional justification for the use of the Bühlmann approach.

Learning objectives

1. Bayesian inference and estimation

2. Prior and posterior pdf

3. Bayesian credibility

4. Conjugate prior distribution

5. Linear exponential distribution

6. Bühlmann credibility versus Bayesian credibility

Bayesian inference and estimation

The classical and Bühlmann credibility models update the prediction for future losses based on recent claim experience and existing prior information.