  In applications of supervised learning applied to medical image segmentation,
the need for large amounts of labeled data typically goes unquestioned. In
particular, in the case of brain anatomy segmentation, hundreds or thousands of
weakly-labeled volumes are often used as training data. In this paper, we first
observe that for many brain structures, a small number of training examples,
(n=9), weakly labeled using Freesurfer 6.0, plus simple data augmentation,
suffice as training data to achieve high performance, achieving an overall mean
Dice coefficient of $0.84 \pm 0.12$ compared to Freesurfer over 28 brain
structures in T1-weighted images of $\approx 4000$ 9-10 year-olds from the
Adolescent Brain Cognitive Development study. We then examine two varieties of
heteroscedastic network as a method for improving classification results. An
existing proposal by Kendall and Gal, which uses Monte-Carlo inference to learn
to predict the variance of each prediction, yields an overall mean Dice of
$0.85 \pm 0.14$ and showed statistically significant improvements over 25 brain
structures. Meanwhile a novel heteroscedastic network which directly learns the
probability that an example has been mislabeled yielded an overall mean Dice of
$0.87 \pm 0.11$ and showed statistically significant improvements over all but
one of the brain structures considered. The loss function associated to this
network can be interpreted as performing a form of learned label smoothing,
where labels are only smoothed where they are judged to be uncertain.
