Marginally constrained nonparametric Bayesian inference through Gaussian processes

Bingjing Tang, Vinayak Rao

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.