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Polya树（PT）先验推广为适用于一组概率测度$G_1,\dots,G_n$的联合先验，发展了一种非参数贝叶斯先验，用于处理事件时间结局的元分析。在元分析应用中，$G_i$表示特定研究$i$的事件时间分布。该模型通过引入具有相似特征的研究对之间的相关性增强，定义了基于研究特定协变量的回归。通过为每个$G_i$的PT构造中的条件分裂概率引入分层先验，构建了所需的多元PT模型。该分层先验用跨所有研究的相应（logit）分裂概率的高斯过程先验替代了PT构造中分裂概率的独立贝塔先验。高斯过程以研究特定协变量为索引，引入了期望的依赖关系，使相似研究具有更强的相关性。该构造的主要特点在于（条件）共轭后验更新，能够兼容事件时间数据常见的推断汇总统计。该方法的提出源于对癌症免疫治疗研究的元分析。