Loss Models: From Data to Decisions, 4th Edition

15.4 Computational issues

It should be obvious by now that all Bayesian analyses proceed by taking integrals or sums. So at least conceptually it is always possible to do a Bayesian analysis. However, only in rare cases are the integrals or sums easy to do, and that means most Bayesian analyses will require numerical integration. While one-dimensional integrations are easy to do to a high degree of accuracy, multidimensional integrals are much more difficult to approximate, A great deal of effort has been expended with regard to solving this problem. A number of ingenious methods have been developed. Some of them are summarized in Klugman [61]. However, the one that is widely used today is called Markov chain Monte Carlo simulation. A good discussion of this method can be found in [100] and actuarial applications can be found in [20] and [101].

¹ In this section and in any subsequent Bayesian discussions, we reserve f (·) for distributions concerning observations (such as the model and predictive distributions) and π(·) for distributions concerning parameters (such as the prior and posterior distributions). The arguments will usually make it clear which particular distribution is being used. To make matters explicit, we also employ subscripts to enable us to keep track of the random variables.