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Sources seem to suggest that when calculating Brier scores involving right-censored data, one must weigh the otherwise mean square error function with the inverse probability of censoring weights method (via Kaplan–Meier estimator). (Example: https://square.github.io/pysurvival/metrics/brier_score.html)

What is the intuition behind such a weighing step? What happens if I skip this step?

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If you had a perfect survival model and censoring was completely at random, then there wouldn't be a problem in skipping that step. In the real world where those assumptions aren't met, you need to take into account the loss of information due to censoring, the association of censoring status with survival-associated covariates, and the mis-specification of the survival model itself. Otherwise you are at risk of bias in your estimate.

Weighting by inverse probability of censoring can overcome these problems. That's similar to how inverse-probability weighting can help with estimating average treatment effects in non-randomized studies, where problems arise associated with the characteristics of individuals who did or did not receive a treatment. As Hernán and Robins put it in Section 8.4:

... we can think of censoring $C$ as just another treatment. That is, the goal of the analysis is to compute the causal effect of a joint intervention on [treatment] $A$ and $C$. To eliminate selection bias for the effect of treatment $A$, we need to adjust for confounding for the effect of treatment $C$.

Continuing in Section 8.5: The inverse-probability weighting means "each [uncensored] individual ... accounts in the analysis not only for herself, but also for those like her, i.e., with the same values of [covariates] and [treatment], who were [censored]." Although they discuss this issue in the context of estimating a causal effect of treatment, the same principles apply to any model-based estimate like the Brier score.

The weighting described in the linked manual page uses a "conditional survival function of the censoring times calculated using the Kaplan-Meier method," which seems to be the method proposed by Graf et al. If that's the case, however, the censoring model is marginal rather than conditional on covariates. Gerds and Schumacher have shown that a proper model of censoring conditional on covariate values is superior and can help overcome a bias in the estimate of the Brier score inherent in mis-specification of the survival model. That is implemented in the R riskRegression package. The result then is a type of "double robust" estimate that should work if either the survival model or the censoring model is correct, but provides no guarantees if both models are incorrect.

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