We develop a non-parametric Bayesian prior for a family of random probability measures by extending the Polya tree ($PT$) prior to a joint prior for a set of probability measures $G_1,\dots,G_n$, suitable for meta-analysis with event time outcomes. In the application to meta-analysis $G_i$ is the event time distribution specific to study $i$. The proposed model defines a regression on study-specific covariates by introducing increased correlation for any pair of studies with similar characteristics. The desired multivariate $PT$ model is constructed by introducing a hierarchical prior on the conditional splitting probabilities in the $PT$ construction for each of the $G_i$. The hierarchical prior replaces the independent beta priors for the splitting probability in the $PT$ construction with a Gaussian process prior for corresponding (logit) splitting probabilities across all studies. The Gaussian process is indexed by study-specific covariates, introducing the desired dependence with increased correlation for similar studies. The main feature of the proposed construction is (conditionally) conjugate posterior updating with commonly reported inference summaries for event time data. The construction is motivated by a meta-analysis over cancer immunotherapy studies.

翻译：我们通过将波利亚树（$PT$）先验扩展至适用于事件时间结果荟萃分析的概率测度集合$G_1,\dots,G_n$的联合先验，为非参数贝叶斯随机概率测度族开发了一种先验。在荟萃分析应用中，$G_i$表示第$i$项研究特有的事件时间分布。该模型通过引入具有相似特征研究对之间的增强相关性，构建了基于研究特异性协变量的回归模型。所提出的多元$PT$模型通过在各个$G_i$的$PT$结构中引入条件分割概率的层次先验而构建。该层次先验将$PT$结构中独立的分割概率贝塔先验替换为跨所有研究对应（对数优势比）分割概率的高斯过程先验。该高斯过程以研究特异性协变量为索引，为相似研究引入具有增强相关性的期望依赖关系。所构建模型的主要特征在于其（条件）共轭后验更新机制，能够兼容事件时间数据常用的推断汇总形式。该模型的构建动机源于对癌症免疫治疗研究的荟萃分析。