  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.
