Loss Models: From Data to Decisions, 4th Edition

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 f_X|(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,…, X_n)^T and x = (x₁,…, x_n)^T, and are interested in setting a rate to cover X_n+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,…, X_n, X_n+1 are independent (although not necessarily identically distributed).

Let X_j have conditional pf

Note that, if the X_js are identically distributed (conditional on = θ), then does not depend on j. Ideally, we are interested in the conditional distribution of X_n+1 given = θ in order to predict the claims ...