Proper Scoring Rules for Right-Censored Survival Data

arXiv:2606.06393v2 Announce Type: replace Abstract: Proper scoring rules provide a rigorous theoretical basis for the training and evaluation of probabilistic forecasts. In survival analysis, such forecasts describe the distribution of the time until an event occurs. However, this event time is often only partially observed because follow-up may end before the event occurs, resulting in right censoring. We propose a framework for proper scoring of right-censored survival outcomes based on a simple idea: first, map the predictive distribution through the censoring mechanism, then apply the underlying proper score on the induced observed-data law. This yields localized scores for fixed censoring times and marginalized scores when the censoring time is random or only partially observed. The resulting construction recovers familiar right-censored likelihood and IPCW-type criteria within a coherent framework, while also yielding right-censored versions of the CRPS, pinball loss, Brier score, and energy score. We show that the marginalized construction is proper under conditional independent censoring and, for strictly proper base scores, identifies the latent joint CDF on the identifiable region. The same principle also leads to censored engression, a sample-based learning objective for multivariate right-censored survival modeling. In experiments, our scores correctly rank the oracle forecast across several censoring regimes, whereas forecast-dependent plug-in weighted scores can exhibit ranking reversals. Censored engression likewise substantially improves over naive training on censored outcomes.

Abstract:Proper scoring rules provide a rigorous theoretical basis for the training and evaluation of probabilistic forecasts. In survival analysis, such forecasts describe the distribution of the time until an event occurs. However, this event time is often only partially observed because follow-up may end before the event occurs, resulting in right censoring. We propose a framework for proper scoring of right-censored survival outcomes based on a simple idea: first, map the predictive distribution through the censoring mechanism, then apply the underlying proper score on the induced observed-data law. This yields localized scores for fixed censoring times and marginalized scores when the censoring time is random or only partially observed. The resulting construction recovers familiar right-censored likelihood and IPCW-type criteria within a coherent framework, while also yielding right-censored versions of the CRPS, pinball loss, Brier score, and energy score. We show that the marginalized construction is proper under conditional independent censoring and, for strictly proper base scores, identifies the latent joint CDF on the identifiable region. The same principle also leads to censored engression, a sample-based learning objective for multivariate right-censored survival modeling. In experiments, our scores correctly rank the oracle forecast across several censoring regimes, whereas forecast-dependent plug-in weighted scores can exhibit ranking reversals. Censored engression likewise substantially improves over naive training on censored outcomes.

Machine Learning (cs.LG)

arXiv:2606.06393 [cs.LG]

(or arXiv:2606.06393v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2606.06393 Focus to learn more arXiv-issued DOI via DataCite

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Proper Scoring Rules for Right-Censored Survival Data

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