  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.
