Bayesian Empirical Bayes: Simultaneous Inference from Probabilistic Symmetries

This paper introduces Bayesian Empirical Bayes (BEB), a novel approach to empirical Bayes methods that leverages probabilistic symmetries to improve simultaneous inference. It addresses the limitations of classical EB theory, which primarily focuses on i.i.d. latent variables, by extending EB to more complex structures like arrays, spatial processes, and covariates. The method's strength lies in its ability to derive EB methods from symmetry assumptions on the joint distribution of latent variables, leading to scalable algorithms based on variational inference and neural networks. The empirical results, demonstrating superior performance in denoising arrays and spatial data, along with real-world applications in gene expression and air quality analysis, highlight the practical significance of BEB.