Every statistical investigation takes place in a context. Information about what question is to be addressed will suggest what data are needed to give useful answers. Before the data are available, one role for this information is to suggest suitable probability models. There may also be information about the values of unknown parameters, and if this can be expressed as a probability density, an approach to inference based on Bayes' theorem is possible. Many statisticians make the stronger claim that this theorem provides the only entirely consistent basis for inference, and insist on its use.

This chapter outlines some aspects of the Bayesian approach to modelling. We first give an account of basic uses of Bayes' theorem and of the role and construction of prior densities. We then turn to inference, dealing with analogues of confidence intervals, tests, approaches to model criticism, and model uncertainty. Until recently computational difficulties placed realistic Bayesian modelling largely out of reach, but over the last 20 years there has been rapid progress and complex models can now be fitted routinely. Section 11.3 gives an account of Bayesian computation, first of analytical approaches based on integral approximations, and then of Monte Carlo methods. The chapter concludes with brief introductions to hierarchical and empirical Bayesian procedures.