  Within Bayesian nonparametrics, dependent Dirichlet process mixture models
provide a highly flexible approach for conducting inference about the
conditional density function. However, several formulations of this class make
either rather restrictive modelling assumptions or involve intricate algorithms
for posterior inference, thus preventing their widespread use. In response to
these challenges, we present a flexible, versatile, and computationally
tractable model for density regression based on a single-weights dependent
Dirichlet process mixture of normal distributions model for univariate
continuous responses. We assume an additive structure for the mean of each
mixture component and incorporate the effects of continuous covariates through
smooth nonlinear functions. The key components of our modelling approach are
penalised B-splines and their bivariate tensor product extension. Our proposed
method also seamlessly accommodates parametric effects of categorical
covariates, linear effects of continuous covariates, interactions between
categorical and/or continuous covariates, varying coefficient terms, and random
effects, which is why we refer our model as a Dirichlet process mixture of
normal structured additive regression models. A noteworthy feature of our
method is its efficiency in posterior simulation through Gibbs sampling, as
closed-form full conditional distributions for all model parameters are
available. Results from a simulation study demonstrate that our approach
successfully recovers true conditional densities and other regression
functionals in various challenging scenarios. Applications to a toxicology,
disease diagnosis, and agricultural study are provided and further underpin the
broad applicability of our modelling framework. An R package, DDPstar,
implementing the proposed method is publicly available at
https://bitbucket.org/mxrodriguez/ddpstar.
