Bayesian nonparametric inference for random distributions and related functions

In recent years, Bayesian nonparametric inference, both theoretical and com putational, has witnessed considerable advances. However, these advances ha ve not received a full critical and comparative analysis of their scope, im pact and limitations in statistical modelling; many aspects of the theory a nd methods remain a mystery to practitioners and many open questions remain . In this paper, we discuss and illustrate the rich modelling and analytic possibilities that are available to the statistician within the Bayesian no nparametric and/or semiparametric framework.