18.3 The Bayesian methodology
Continue to assume that the distribution of the risk characteristics in the population may be represented by π(θ), and the experience of a particular policyholder with risk parameter θ arises from the conditional distribution fX|(x|θ) of claims or losses given θ.
We now return to the problem introduced in Section 17.2. That is, for a particular policyholder, we have observed X = x, where X = (X1,…, Xn)T and x = (x1,…, xn)T, and are interested in setting a rate to cover Xn+1. We assume that the risk parameter associated with the policyholder is θ (which is unknown). Furthermore, the experience of the policyholder corresponding to different exposure periods is assumed to be independent. In statistical terms, conditional on θ, the claims or losses X1,…, Xn, Xn+1 are independent (although not necessarily identically distributed).
Let Xj have conditional pf
Note that, if the Xjs are identically distributed (conditional on = θ), then does not depend on j. Ideally, we are interested in the conditional distribution of Xn+1 given = θ in order to predict the claims ...