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.

翻译：分段常数先验在生存分析的贝叶斯Cox比例风险模型中广泛应用。尽管该方法应用普遍，但其贝叶斯方法的大样本性质尚未得到充分理解。本文为这一设定下的后验分布提供了统一理论，不要求先验具有共轭性。首先，我们针对未知风险函数上广泛的直方图先验类推导了收缩率结果，并通过伯恩斯坦-冯·米塞斯定理证明了后验风险线性泛函的渐近正态性。其次，利用近年来发展的多尺度技术，我们推导了累积风险函数和生存函数的泛函极限结果。本文还研究了贝叶斯可信集在频率学派框架下的覆盖性质：我们证明某些易于计算的生存函数可信带具有最优的频率学派置信带性质。仿真研究验证了这些理论预测，尤其在有限样本中表现出优异特性。我们的研究结果表明，贝叶斯方法能够为实际应用中的系数估计和生存函数可信带计算提供简便解决方案。