A Bayesian treatment of the proportional hazard (PH) model is revisited using Dirichlet process and gamma process priors for the baseline survival and cumulative hazard functions respectively. Such priors, due to their discrete support, conflict with the absolutely continuous nature of survival responses expressed through the hazard function structure that defines the PH model. We resolve this conflict through the use of a proposed $\epsilon$-grid likelihood approach which we apply to the PH model to consider marginal inference for the regression parameter $\beta$ using expansions as $\epsilon \downarrow 0$. Using Dirichlet process priors, the $\epsilon$-grid likelihood approach leads to a new explicit marginal posterior density expansion for $\beta$ which accommodates the most general setting with arbitrary ties and right censoring. In the previously considered case of gamma process priors, the approach extends the results of Kalbfleisch (1978) to deal with arbitrary arrangements of ties and also provides a rigorous justification for his posterior expressions. As in Kalbfleisch (1978), we show that the leading terms of both expansions approximate Cox’s partial likelihood when there are no ties and the process priors are diffuse. The $\epsilon$-grid likelihood approach is similar in concept to the grouped data likelihood approach of Sinha et al. (2003) but differs in the likelihood approximation. Whereas our expansions are Poincaré expansions (with relative error $O\left(\epsilon \right)$ as $\epsilon \downarrow 0$), those of Sinha et al. (2003) are stochastic expansions and not proper Poincaré expansion (as we show), so that they lack relative error $O\left(\epsilon \right)$. Using a flat improper prior for $\beta$, the two marginal posterior expressions for $\beta$ are shown to be integrable for general arrangements of ties and censoring under weak conditions on the design and failures.

分别使用 Dirichlet 过程和伽马过程先验对比例风险 (PH) 模型进行贝叶斯处理，分别用于基线生存和累积风险函数。由于它们的离散支持，这些先验与通过定义 PH 模型的风险函数结构表达的生存响应的绝对连续性相冲突。我们通过使用建议的$\epsilon$- 我们应用于 PH 模型以考虑回归参数的边际推断的网格似然方法 $\beta$ 使用扩展作为 $\epsilon \downarrow 0$. 使用 Dirichlet 过程先验，$\epsilon$- 网格似然方法导致新的显式边际后验密度扩展 $\beta$它适用于具有任意关系和右删失的最一般设置。在先前考虑的伽马过程先验情况中，该方法扩展了 Kalbfleisch (1978) 的结果以处理任意的关系安排，并为其后验表达式提供了严格的证明。与 Kalbfleisch (1978) 一样，我们表明，当没有联系且过程先验是扩散的时，两个扩展的主要项都近似于 Cox 的部分似然。这$\epsilon$-grid 似然方法在概念上类似于 Sinha 等人的分组数据似然方法。(2003) 但在似然近似方面有所不同。而我们的展开式是庞加莱展开式（相对误差$哦\left(\epsilon \right)$ 作为 $\epsilon \downarrow 0$)，那些辛哈等人。(2003) 是随机展开式而不是适当的庞加莱展开式（如我们所示），因此它们缺乏相对误差$哦\left(\epsilon \right)$. 使用平坦的不适当的先验$\beta$，两个边际后验表达式为 $\beta$ 在设计和故障的弱条件下，对于连接和审查的一般安排来说，它们是可集成的。