  Nonparametric Bayesian models are used routinely as flexible and powerful
models of complex data. Many times, a statistician may have additional
informative beliefs about data distribution of interest, e.g., its mean or
subset components, that is not part of, or even compatible with, the
nonparametric prior. An important challenge is then to incorporate this partial
prior belief into nonparametric Bayesian models. In this paper, we are
motivated by settings where practitioners have additional distributional
information about a subset of the coordinates of the observations being
modeled. Our approach links this problem to that of conditional density
modeling. Our main idea is a novel constrained Bayesian model, based on a
perturbation of a parametric distribution with a transformed Gaussian process
prior on the perturbation function. We also develop a corresponding posterior
sampling method based on data augmentation. We illustrate the efficacy of our
proposed constrained nonparametric Bayesian model in a variety of real-world
scenarios including modeling environmental and earthquake data.
