  We present a novel Bayesian nonparametric regression model for covariates X
and continuous, real response variable Y. The model is parametrized in terms of
marginal distributions for Y and X and a regression function which tunes the
stochastic ordering of the conditional distributions F(y|x). By adopting an
approximate composite likelihood approach, we show that the resulting posterior
inference can be decoupled for the separate components of the model. This
procedure can scale to very large datasets and allows for the use of standard,
existing, software from Bayesian nonparametric density estimation and
Plackett-Luce ranking estimation to be applied. As an illustration, we show an
application of our approach to a US Census dataset, with over 1,300,000 data
points and more than 100 covariates.
