  Deep models trained through maximum likelihood have achieved state-of-the-art
results for survival analysis. Despite this training scheme, practitioners
evaluate models under other criteria, such as binary classification losses at a
chosen set of time horizons, e.g. Brier score (BS) and Bernoulli log likelihood
(BLL). Models trained with maximum likelihood may have poor BS or BLL since
maximum likelihood does not directly optimize these criteria. Directly
optimizing criteria like BS requires inverse-weighting by the censoring
distribution. However, estimating the censoring model under these metrics
requires inverse-weighting by the failure distribution. The objective for each
model requires the other, but neither are known. To resolve this dilemma, we
introduce Inverse-Weighted Survival Games. In these games, objectives for each
model are built from re-weighted estimates featuring the other model, where the
latter is held fixed during training. When the loss is proper, we show that the
games always have the true failure and censoring distributions as a stationary
point. This means models in the game do not leave the correct distributions
once reached. We construct one case where this stationary point is unique. We
show that these games optimize BS on simulations and then apply these
principles on real world cancer and critically-ill patient data.
