Scalable Bayesian transport maps for high-dimensional non-Gaussian spatial fields

TL;DR

This paper introduces a scalable Bayesian approach using Gaussian process-based transport maps for modeling high-dimensional, non-Gaussian spatial fields, enabling uncertainty quantification and efficient inference from limited data.

Contribution

It develops a Bayesian nonparametric framework for transport maps with Gaussian process priors, allowing scalable, flexible, and uncertainty-aware inference of complex spatial distributions.

Findings

Accurate modeling of non-Gaussian climate-model outputs.

Scalable inference in high-dimensional spatial fields.

Effective uncertainty quantification of transport maps.

Abstract

A multivariate distribution can be described by a triangular transport map from the target distribution to a simple reference distribution. We propose Bayesian nonparametric inference on the transport map by modeling its components using Gaussian processes. This enables regularization and uncertainty quantification of the map estimation, while still resulting in a closed-form and invertible posterior map. We then focus on inferring the distribution of a nonstationary spatial field from a small number of replicates. We develop specific transport-map priors that are highly flexible and are motivated by the behavior of a large class of stochastic processes. Our approach is scalable to high-dimensional distributions due to data-dependent sparsity and parallel computations. We also discuss extensions, including Dirichlet process mixtures for flexible marginals. We present numerical results to demonstrate the accuracy, scalability, and usefulness of our methods, including statistical emulation of non-Gaussian climate-model output.