Abstract
Piecewise constant priors are routinely used in the Bayesian Cox proportional hazards model for survival analysis. Despite its popularity, large sample properties of this Bayesian method are not yet well understood. This work provides a unified theory for posterior distributions in this setting, not requiring the priors to be conjugate. We first derive contraction rate results for wide classes of histogram priors on the unknown hazard function and prove asymptotic normality of linear functionals of the posterior hazard in the form of Bernstein–von Mises theorems. Second, using recently developed multiscale techniques, we derive functional limiting results for the cumulative hazard and survival function. Frequentist coverage properties of Bayesian credible sets are investigated: we prove that certain easily computable credible bands for the survival function are optimal frequentist confidence bands. We conduct simulation studies that confirm these predictions, with an excellent behavior particularly in finite samples. Our results suggest that the Bayesian approach can provide an easy solution to obtain both the coefficients estimate and the credible bands for survival function in practice.
Funding Statement
The first author gratefully acknowledges the support from the Fondation Sciences Mathématiques de Paris postdoctoral fellowship. The second author gratefully acknowledges support from the Institut Universitaire de France (IUF) and from the ANR grant ANR-17-CE40-0001 (BASICS).
Acknowledgements
The authors would like to thank Stéphanie van der Pas for helpful discussions with the code for simulation studies. The authors would also like to thank the Associate Editor and two referees for insightful comments.
Citation
Bo Y.-C. Ning. Ismaël Castillo. "Bayesian multiscale analysis of the Cox model." Bernoulli 30 (2) 1525 - 1554, May 2024. https://doi.org/10.3150/23-BEJ1643
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