Marginally Constrained Nonparametric Bayesian Inference through Gaussian Processes

TL;DR

This paper introduces a novel constrained nonparametric Bayesian approach using Gaussian processes to incorporate partial prior beliefs about data distribution, demonstrated through real-world environmental and earthquake data modeling.

Contribution

It develops a new Bayesian model that perturbs parametric distributions with Gaussian process priors, enabling the integration of partial prior information into nonparametric inference.

Findings

Effective in modeling environmental data

Improves inference with partial prior knowledge

Demonstrated on earthquake data

Abstract

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.