Generalized Bayesian Inference using the Bayesian Bootstrap for Survival Models
Survival inference often requires uncertainty quantification under censoring and limited sample sizes, while prior information may be available but difficult to incorporate without specifying a full likelihood. Likelihood-based Bayesian survival methods provide prior-informed inference but may be sensitive to distributional assumptions and computationally demanding for complex survival models. The Bayesian bootstrap provides a likelihood-free, nonparametric approach to uncertainty quantification, but by itself does not provide a general mechanism for incorporating priors on model parameters. We propose a general Bayesian method for survival models that combines the generalized Bayesian (Gibbs) updating with Bayesian bootstrap. The Bayesian bootstrap generates a distribution over a target survival estimator through Dirichlet weights, while generalized Bayesian updating incorporates parameter-specific prior information through a loss function. The resulting posterior provides a flexible alternative to likelihood-based Bayesian inference and can be applied to a broad class of survival estimators. We develop the method for the Cox proportional hazards model and obtain posterior inference for regression coefficients and hazard ratios. Simulation studies demonstrate uncertainty quantification and prior-data learning across varying sample sizes. An application to right-censored survival data illustrates its practical utility. The methodology is implemented in the open-source \texttt{R} package \texttt{BayesBoots}.