Fréchet regression provides a versatile framework for modeling responses in metric spaces with Euclidean predictors, yet current methodologies rely almost exclusively on frequentist approaches. We propose a Bayesian framework for Fréchet regression that offers a principled way of incorporating prior information into nonlinear global Fréchet regression. By targeting a novel Fréchet Bayes rule, we reduce the object-valued regression problem to a collection of tractable scalar regression tasks. Our approach allows for a controlled interpolation between the prior and the data-driven frequentist estimate, facilitating effective shrinkage toward informed values. While initially derived under Gaussian assumptions, we demonstrate that our framework is robust to model misspecification by establishing its validity under moment conditions via weak conditional expectations. The numerical properties of the proposed methodology are demonstrated in simulation studies and an application to microbiome compositional data, where we show that leveraging an auxiliary cohort to inform the prior significantly enhances predictive performance in a targeted, small-scale study

翻译：弗雷歇回归为度量空间中响应变量与欧几里得预测变量的建模提供了通用框架，然而现有方法几乎完全依赖频率学派范式。本文提出贝叶斯弗雷歇回归框架，通过将先验信息系统性地融入非线性全局弗雷歇回归，实现了对传统方法的理论升华。通过定义新型弗雷歇贝叶斯规则，我们将对象值回归问题转化为一系列可解的标量回归任务。该方法可实现先验估计与数据驱动的频率学派估计之间的受控插值，从而促进对先验信息的有效收缩。虽最初在高斯假设下推导，但通过弱条件期望证明其在矩条件下的有效性，证实该框架对模型误设具有稳健性。仿真实验与微生物组组成数据分析表明，利用辅助队列构建先验信息可显著提升小规模靶向研究的预测性能。