  We develop Bayesian models for density regression with emphasis on discrete
outcomes. The problem of density regression is approached by considering
methods for multivariate density estimation of mixed scale variables, and
obtaining conditional densities from the multivariate ones. The approach to
multivariate mixed scale outcome density estimation that we describe represents
discrete variables, either responses or covariates, as discretised versions of
continuous latent variables. We present and compare several models for
obtaining these thresholds in the challenging context of count data analysis
where the response may be over- and/or under-dispersed in some of the regions
of the covariate space. We utilise a nonparametric mixture of multivariate
Gaussians to model the directly observed and the latent continuous variables.
The paper presents a Markov chain Monte Carlo algorithm for posterior sampling,
sufficient conditions for weak consistency, and illustrations on density, mean
and quantile regression utilizing simulated and real datasets.
