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.

翻译：从目标分布到简单参考分布的三角运输图可以描述多变量分布。 我们建议使用高山进程对运输图的部件进行模型化,以此对运输图的部件进行贝叶斯非参数性推断。 这样可以对地图估计进行正规化和不确定性量化,同时仍然产生一个封闭式和不可翻转的外表地图。 然后我们集中从少量的复制品中推断出非静止空间场的分布。 我们开发了非常灵活并受大规模随机过程行为驱动的特定的运输图前科。 我们的方法由于依赖数据的宽度和平行计算,可以对高维分布进行伸缩。 我们还讨论扩展,包括灵活边缘的Drichlet工艺混合物。 我们提出数字结果,以显示我们方法的准确性、可伸缩性和有用性,包括非加西气候模型产出的统计模拟。