We propose two Bayesian bootstrap extensions, the binomial and Poisson forms, for proportional hazards models. The binomial form Bayesian bootstrap is the limit of the posterior distribution with a beta process prior as the amount of the prior information vanishes, and thus can be considered as a default nonparametric Bayesian analysis. It is also the same as Lo's Bayesian bootstrap for censored data when covariates are absent. The Poisson form Bayesian bootstrap is equivalent to the Bayesian analysis with Cox's profile likelihood. When the baseline distribution is discrete, thus when the data set has many ties, simulation study suggests that the binomial form Bayesian bootstrap performs better than standard frequentist procedures in the frequentist sense. An advantage of the proposed Bayesian bootstrap procedures over the standard Bayesian analysis is conceptual and computational simplicity. Finally, it is shown that both Bayesian bootstrap posteriors are asymptotically equivalent to the sampling distribution of the maximum likelihood estimator.
"Bayesian bootstrap for proportional hazards models." Ann. Statist. 31 (6) 1905 - 1922, December 2003. https://doi.org/10.1214/aos/1074290331