  Survival analysis models the distribution of time until an event of interest,
such as discharge from the hospital or admission to the ICU. When a model's
predicted number of events within any time interval is similar to the observed
number, it is called well-calibrated. A survival model's calibration can be
measured using, for instance, distributional calibration (D-CALIBRATION)
[Haider et al., 2020] which computes the squared difference between the
observed and predicted number of events within different time intervals.
Classically, calibration is addressed in post-training analysis. We develop
explicit calibration (X-CAL), which turns D-CALIBRATION into a differentiable
objective that can be used in survival modeling alongside maximum likelihood
estimation and other objectives. X-CAL allows practitioners to directly
optimize calibration and strike a desired balance between predictive power and
calibration. In our experiments, we fit a variety of shallow and deep models on
simulated data, a survival dataset based on MNIST, on length-of-stay prediction
using MIMIC-III data, and on brain cancer data from The Cancer Genome Atlas. We
show that the models we study can be miscalibrated. We give experimental
evidence on these datasets that X-CAL improves D-CALIBRATION without a large
decrease in concordance or likelihood.
