Modeling Bayesian Transport Map Uncertainty for Non-Gaussian Spatial Data via Laplace Approximation

Music City Center 

Transport maps can be used to describe non-Gaussian multivariate distributions relative to a simple reference distribution, usually Gaussian. Previous work in this area focused on modeling transport maps using Gaussian processes, and computational limitations have led practitioners to focus on learning map parameters via stochastic gradient methods. We extend this idea by employing a Laplace approximation to the posterior distribution of transport map parameters. We first discuss the characteristics of the Laplace approximation in the transport map setting, then explore how capturing and quantifying uncertainty in transport map parameters affects the model's ability to learn the non-Gaussian target distribution. We then compare our new model's performance in learning the distribution of a potentially nonstationary spatial field to established methods using various metrics. Finally, we contrast the Laplace approximation with various other approximation and uncertainty quantification methods.

Gaussian process

Generative modeling

Laplace approximation

Uncertainty quantification 

Section on Statistics and the Environment