| Summary: | This thesis is about insurance models and aspects of uncertainty pertaining to such models. The models we consider are insurance credibility models, arising from the need for accurate rate making based on past experience of claims in some portfolio of insurance policies. Classical credibility modelling is concerned with the use of a linear estimate to approximate the risk premium and was first studied by American actuaries at the start of the 20<sup>th</sup> century. In the Bayesian paradigm the credibility premium is the optimal linear premium since it minimises the expected square loss based on current information. Here we focus on estimating the risk premium without using the linear estimator since the linear estimate is known to be an exact expression only in certain restricted cases such as the linear exponential family. Markov chain Monte Carlo (MCMC) has become a standard tool in statistical analysis. In this thesis we show how it can be used in a Bayesian setting applied to insurance credibility theory. Using MCMC methods, we can compute the premium to cover future risks to any degree of accuracy required by simulating directly from the posterior distribution of the unknown model risk parameters and then averaging the risk premium against this distribution. This is illustrated for a special case. We then consider the problem of model uncertainty and model selection in general credibility modelling. This is necessary especially when there are several competing models which seem to adequately describe the data. Most of our model selection techniques are based on the reversible jump MCMC algorithm of Green (1995, Biometrika). Recently Brooks et al. (2003, JRSSB) have proposed several implementational improvements for the vanilla reversible jump algorithm. In this thesis we apply these methods to various model selection problems in insurance credibility theory.
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