Nonparametric f-Modeling for Empirical Bayes Inference with Unequal and Unknown Variances

2026-04-23

AI Summary

This work addresses the challenge of unknown and heterogeneous observation variances in heteroscedastic normal mean models by proposing a nonparametric empirical Bayes approach that enables full posterior inference without specifying or estimating a prior distribution. By developing a generalized Tweedie-type identity, the classical f-modeling framework is extended to accommodate variance heterogeneity. Bayesian estimators are expressed in terms of the joint marginal density and its partial derivatives, and the complete posterior distribution is recovered via the moment-generating function. The method achieves accurate shrinkage estimation in both simulated and real data, while supporting reliable uncertainty quantification, point estimation, and hypothesis testing. Notably, it represents the first approach within the f-modeling paradigm to achieve prior-free full posterior inference.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Bayesian Learning
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Empirical Bayes methods are widely used for large-scale inference, yet most classical approaches assume homoscedastic observations and focus primarily on posterior mean estimation. We develop a nonparametric empirical Bayes framework for the heteroscedastic normal means problem with unequal and unknown variances. Our first contribution is a generalized Tweedie-type identity that expresses the Bayes estimator entirely in terms of the joint marginal density of the observed statistics and its partial derivatives, extending the classical Tweedie's formula to settings with heterogeneous and unknown variances. Our second contribution is to introduce a moment-generating-function representation that enables recovery of the full posterior distribution within the f-modeling paradigm without specifying or estimating the prior distribution. The resulting method provides a unified framework for point estimation, uncertainty quantification, and hypothesis testing while accommodating arbitrary dependence between means and variances. Simulation studies and real-data analysis demonstrate that the proposed approach achieves accurate shrinkage estimation and reliable posterior inference in heterogeneous data environments.

Problem

• empirical Bayes
• heteroscedasticity
• unknown variances
• nonparametric f-modeling
• normal means problem

Innovation

• nonparametric empirical Bayes
• heteroscedasticity
• Tweedie's formula
• posterior inference
• f-modeling