  Machine learning driven medical image segmentation has become standard in
medical image analysis. However, deep learning models are prone to
overconfident predictions. This has led to a renewed focus on calibrated
predictions in the medical imaging and broader machine learning communities.
Calibrated predictions are estimates of the probability of a label that
correspond to the true expected value of the label conditioned on the
confidence. Such calibrated predictions have utility in a range of medical
imaging applications, including surgical planning under uncertainty and active
learning systems. At the same time it is often an accurate volume measurement
that is of real importance for many medical applications. This work
investigates the relationship between model calibration and volume estimation.
We demonstrate both mathematically and empirically that if the predictor is
calibrated per image, we can obtain the correct volume by taking an expectation
of the probability scores per pixel/voxel of the image. Furthermore, we show
that convex combinations of calibrated classifiers preserve volume estimation,
but do not preserve calibration. Therefore, we conclude that having a
calibrated predictor is a sufficient, but not necessary condition for obtaining
an unbiased estimate of the volume. We validate our theoretical findings
empirically on a collection of 18 different (calibrated) training strategies on
the tasks of glioma volume estimation on BraTS 2018, and ischemic stroke lesion
volume estimation on ISLES 2018 datasets.
