Loss Models: From Data to Decisions, 4th Edition

15.1 Definitions and Bayes’ Theorem

Definition 15.1 The prior distribution is a probability distribution over the space of possible parameter values. It is denoted π(θ) and represents our opinion concerning the relative chances that various values of θ are the true value of the parameter.

As before, the parameter θ may be scalar or vector valued. Determination of the prior distribution has always been one of the barriers to the widespread acceptance of Bayesian methods. It is almost certainly the case that your experience has provided some insights about possible parameter values before the first data point has been observed. (If you have no such opinions, perhaps the wisdom of the person who assigned this task to you should be questioned.) The difficulty is translating this knowledge into a probability distribution. An excellent discussion about prior distributions and the foundations of Bayesian analysis can be found in Lindley [68], and for a discussion about issues surrounding the choice of Bayesian versus frequentist methods, see Efron [30]. The book by Klugman [61] contains more detail on the Bayesian approach along with several actuarial applications. More recent articles applying Bayesian methods to actuarial problems include [24], [76], [77], [83], [101], [114], and [122]. A good source for a thorough mathematical treatment of Bayesian methods is the text by Berger [13]. In recent years, many advancements in Bayesian calculations have occurred. A good resource is [21]. ...