Loss Models: From Data to Decisions, 4th Edition

19.1 Introduction

In Chapter 18, a modeling methodology is proposed that suggests the use of either the Bayesian or credibility premium as a way to incorporate past data into the prospective rate. There is a practical problem associated with the use of these models that has not yet been addressed.

In the examples seen so far, we are able to obtain numerical values for the quantities of interest because the input distributions f_Xj|(x_j|θ) and π(θ) are assumed to be known. These examples, while useful for illustration of the methodology, can hardly be expected to accurately represent the business of an insurance portfolio. More practical models of necessity involve the use of parameters that must be chosen to ensure a close agreement between the model and reality. Examples of this include: the Poisson–gamma model (Example 18.1), where the gamma parameters α and β need to be selected, or the Bühlmann or Bühlmann–Straub parameters μ, v, and a. Assignment of numerical values to the Bayesian or credibility premium requires that these parameters be replaced by numerical values.

In general, the unknown parameters are those associated with the structure density π(θ), and, hence we refer to these as structural parameters. The terminology we use follows the Bayesian framework of the previous chapter. Strictly speaking, in the Bayesian context all structural parameters are assumed known and there ...