Distribution regression has recently attracted
much interest as a generic solution to the problem of supervised learning where labels are available at the group level, rather than at the individual level. Current approaches, however, do not
propagate the uncertainty in observations due to
sampling variability in the groups. This effectively assumes that small and large groups are
estimated equally well, and should have equal
weight in the final regression. We account for
this uncertainty with a Bayesian distribution regression formalism, improving the robustness
and performance of the model when group sizes
vary. We frame our models in a neural network
style, allowing for simple MAP inference using
backpropagation to learn the parameters, as well
as MCMC-based inference which can fully propagate uncertainty. We demonstrate our approach
on illustrative toy datasets, as well as on a challenging problem of predicting age from images.