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Within Bayesian nonparametrics, dependent Dirichlet process mixture modelsprovide a highly flexible approach for conducting inference about theconditional density function. However, several formulations of this class makeeither rather restrictive modelling assumptions or involve intricate algorithmsfor posterior inference, thus preventing their widespread use. In response tothese challenges, we present a flexible, versatile, and computationallytractable model for density regression based on a single-weights dependentDirichlet process mixture of normal distributions model for univariatecontinuous responses. We assume an additive structure for the mean of eachmixture component and incorporate the effects of continuous covariates throughsmooth nonlinear functions. The key components of our modelling approach arepenalised B-splines and their bivariate tensor product extension. Our proposedmethod also seamlessly accommodates parametric effects of categoricalcovariates, linear effects of continuous covariates, interactions betweencategorical and/or continuous covariates, varying coefficient terms, and randomeffects, which is why we refer our model as a Dirichlet process mixture ofnormal structured additive regression models. A noteworthy feature of ourmethod is its efficiency in posterior simulation through Gibbs sampling, asclosed-form full conditional distributions for all model parameters areavailable. Results from a simulation study demonstrate that our approachsuccessfully recovers true conditional densities and other regressionfunctionals in various challenging scenarios. Applications to a toxicology,disease diagnosis, and agricultural study are provided and further underpin thebroad applicability of our modelling framework. An R package, \texttt{DDPstar},implementing the proposed method is publicly available at\url{https://bitbucket.org/mxrodriguez/ddpstar}.

Density regression via Dirichlet process mixtures of normal structured additive regression models