# Bayesian survival analysis

Survival analysis delivers answers to questions of the form "How long until...?". For example, "How long until a patient needs dialysis after diagnosis with diabetes?"

With Ismaël Castillo, I investigated whether Bayesian survival analysis can result in reliable quantification of the uncertainty in survival estimates. We showed that the answer is "yes" by proving Bernstein-von Mises results in the supremum norm, and made software (see Software) available to apply the methods we studied.

## Summary

The survival function indicates for timepoints t the probability that the event of interest occurs after time t. A popular method to estimate the survival function is the Kaplan-Meier estimator. The uncertainty in the estimate is then typically indicated by plotting pointwise confidence intervals. The 'pointwise' part is crucial, this means the confidence interval only indicates the uncertainty at a particular point in time. In contrast, a confidence band indicates the uncertainty for the entire survival function.

In this project, we investigated the theoretical properties of piecewise constant priors on the hazard in the right-censoring survival model. We studied both independent priors, where the independence refers to the relationship between individual pieces, and dependent priors with a Markov structure. We derived posterior limiting distributions for linear functionals of the hazard, and then for ‘many’ functionals simultaneously in appropriate multiscale spaces. As an application, we derived Bernstein–von Mises theorems for the cumulative hazard and survival functions, which lead to asymptotically efficient confidence bands for these quantities. Further, we showed optimal posterior contraction rates for the hazard in terms of the supremum norm.

The samplers we used in the paper are available as an R software package called BayesSurvival, see Software.

## Related papers

Castillo, I., & van der Pas, S. (2021). Multiscale Bayesian survival analysis. The Annals of Statistics, 49(6), 3559-3582. [link]