Scalable Bayesian Transport Maps for High-Dimensional Non-Gaussian Spatial Fields

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 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. Supplementary materials for this article are available online.

keyword:

• Climate-model emulation
• Dirichlet process mixture
• Gaussian process
• Generative modeling
• Maximin ordering
• Nonstationarity

Supplementary Materials

Appendices A–G contain proofs, a discussion of conditional near-Gaussianity for quasiquadratic loglikelihoods, details on the Gibbs sampler for Dirichlet process mixture model, and additional numerical results and comparisons.

Disclosure Statement

The authors report there are no competing interests to declare.

Acknowledgments

We would like to thank Joe Guinness and several reviewers for helpful comments. We are especially grateful to Jian Cao, who wrote a Python implementation, produced timing results, and obtained GAN and VAE results, and to Trevor Harris, who created the VAE implementation for our numerical comparisons.

Additional information

Funding

Katzfuss was partially supported by National Science Foundation (NSF) grants DMS–1654083, DMS–1953005, and CCF–1934904, and by NASA’s Advanced Information Systems Technology Program (AIST-21). Schäfer gratefully acknowledges support by the Air Force Office of Scientific Research under award number FA9550-18-1-0271, and the Office of Naval Research under award N00014-18-1-2363.