Inverse-Weighted Survival Games
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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, estimation of which itself also requires inverse-weighted by the failure distribution. But neither are known. To resolve this dilemma, we introduce Inverse-Weighted Survival Games to train both failure and censoring models with respect to criteria such as BS or BLL. In these games, objectives for each model are built from re-weighted estimates featuring the other model, where the re-weighting model 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.
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