2 The Dirichlet process , related priors and posterior asymptotics

Here we review the role of the Dirichlet process and related prior distribtions in nonparametric Bayesian inference. We discuss construction and various properties of the Dirichlet process. We then review the asymptotic properties of posterior distributions. Starting with the definition of posterior consistency and examples of inconsistency, we discuss general theorems which lead to consistency. We then describe the method of calculating posterior convergence rates and briefly outline how such rates can be computed in nonparametric examples. We also discuss the issue of posterior rate adaptation, Bayes factor consistency in model selection and Bernshtěın–von Mises type theorems for nonparametric problems.