  Standard regression approaches assume that some finite number of the response
distribution characteristics, such as location and scale, change as a
(parametric or nonparametric) function of predictors. However, it is not always
appropriate to assume a location/scale representation, where the error
distribution has unchanging shape over the predictor space. In fact, it often
happens in applied research that the distribution of responses under study
changes with predictors in ways that cannot be reasonably represented by a
finite dimensional functional form. This can seriously affect the answers to
the scientific questions of interest, and therefore more general approaches are
indeed needed. This gives rise to the study of fully nonparametric regression
models. We review some of the main Bayesian approaches that have been employed
to define probability models where the complete response distribution may vary
flexibly with predictors. We focus on developments based on modifications of
the Dirichlet process, historically termed dependent Dirichlet processes, and
some of the extensions that have been proposed to tackle this general problem
using nonparametric approaches.
