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Nonparametric Bayesian models are used routinely as flexible and powerfulmodels of complex data. Many times, a statistician may have additionalinformative beliefs about data distribution of interest, e.g., its mean orsubset components, that is not part of, or even compatible with, thenonparametric prior. An important challenge is then to incorporate this partialprior belief into nonparametric Bayesian models. In this paper, we aremotivated by settings where practitioners have additional distributionalinformation about a subset of the coordinates of the observations beingmodeled. Our approach links this problem to that of conditional densitymodeling. Our main idea is a novel constrained Bayesian model, based on aperturbation of a parametric distribution with a transformed Gaussian processprior on the perturbation function. We also develop a corresponding posteriorsampling method based on data augmentation. We illustrate the efficacy of ourproposed constrained nonparametric Bayesian model in a variety of real-worldscenarios including modeling environmental and earthquake data.

Marginally Constrained Nonparametric Bayesian Inference through Gaussian Processes