Marginally constrained nonparametric Bayesian inference through Gaussian processes

Abstract

Nonparametric Bayesian models are used routinely as flexible and powerful models of complex data. In many situations, an applied scientist may have additional informative beliefs about the data distribution of interest, for instance, the distribution of its mean or a subset components. This often will not be 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 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.