Bayesian Global Fréchet Regression via Weak Conditional Expectations

2026-06-05

AI Summary

Existing Fréchet regression approaches predominantly rely on frequentist frameworks, which struggle to effectively incorporate prior information. This work proposes the first Bayesian global Fréchet regression method, leveraging a novel Fréchet Bayesian rule to decompose object-valued regression into multiple scalar tasks. By integrating the theory of weak conditional expectation, the approach ensures robustness without requiring Gaussian assumptions. The method enables the construction of informative priors from auxiliary cohorts, allowing for controllable fusion of prior knowledge with data-driven estimation—substantially enhancing predictive performance in small-sample settings. Simulations and real-world microbiome data analyses demonstrate its efficacy, particularly when target samples are limited: incorporating priors from external cohorts markedly improves prediction accuracy.

Abstract

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

Problem

• Fréchet regression
• Bayesian framework
• metric space
• prior information
• object-valued response

Innovation

• Bayesian Fréchet regression
• weak conditional expectations
• object-valued regression
• prior-data interpolation
• model robustness