  We propose a parameter efficient Bayesian layer for hierarchical
convolutional Gaussian Processes that incorporates Gaussian Processes operating
in Wasserstein-2 space to reliably propagate uncertainty. This directly
replaces convolving Gaussian Processes with a distance-preserving affine
operator on distributions. Our experiments on brain tissue-segmentation show
that the resulting architecture approaches the performance of well-established
deterministic segmentation algorithms (U-Net), which has never been achieved
with previous hierarchical Gaussian Processes. Moreover, by applying the same
segmentation model to out-of-distribution data (i.e., images with pathology
such as brain tumors), we show that our uncertainty estimates result in
out-of-distribution detection that outperforms the capabilities of previous
Bayesian networks and reconstruction-based approaches that learn normative
distributions.
