Online Bayesian Inference for Cox Proportional Hazards Model

In this article, we develop an online Bayesian inferential method for the Cox proportional hazards model with right-censored data. The proposed method is designed to analyze datasets where mini-batches of the entire data are sequentially available. As each mini-batch arrives, the method updates the current prior to the posterior distribution, which is then used as the prior for the next step. Each update consists of two steps: updating the marginal posterior of the regression coefficients using the partial likelihood and updating the remaining conditional posterior using the Poisson form Bayesian bootstrap likelihood. Our method is capable of inferring both the regression coefficients and the baseline cumulative hazard function. To the best of our knowledge, this is the first online Bayesian method for the Cox proportional hazards model. Through numerical experiments and real data analysis, we demonstrate that the proposed method outperforms existing frequentist online methods and is comparable to batch learning on the entire dataset, even when the size of mini-batches is moderately small. Supplementary materials for this article are available online.

本文针对带右删失数据（right-censored data）的考克斯比例风险模型（Cox proportional hazards model），提出了一种在线贝叶斯推断方法（online Bayesian inferential method）。所提方法旨在针对全量数据以小批量（mini-batches）形式依次可用的数据集开展分析。每当一批小批量数据抵达时，该方法会将当前先验分布更新为后验分布，而后将该后验分布用作下一步的先验。每一次更新包含两个步骤：一是利用偏似然（partial likelihood）更新回归系数的边缘后验分布，二是借助泊松形式贝叶斯自举似然（Poisson form Bayesian bootstrap likelihood）更新其余条件后验分布。本方法可同时推断回归系数与基线累积风险函数（baseline cumulative hazard function）。据我们所知，这是首个针对考克斯比例风险模型的在线贝叶斯方法。通过数值实验与真实数据分析，本文证明所提方法优于现有频率学派在线方法（frequentist online methods），且即便小批量规模适中偏小，其性能也可与全量数据集上的批量学习（batch learning）相媲美。本文的补充材料可在线获取。