On Quantification of Borrowing of Information in Hierarchical Bayesian Models

2025-09-21

AI Summary

This paper investigates the “information borrowing” mechanism driven by shared hyperparameters in hierarchical Bayesian models and its impact on posterior inference performance. Method: Focusing on mixed-effects models, we propose an integrated risk measure grounded in the true data-generating distribution and develop a non-asymptotic theoretical framework to quantify information borrowing efficiency under varying prior depths. Contribution/Results: We prove that when random effects exhibit a compound-symmetric structure—especially with strong between-group correlation—the Bayesian posterior mean estimator from a deeply nested hierarchical model strictly dominates that from any shallower nested submodel; we further derive necessary and sufficient conditions for this dominance. The result extends to perturbed correlation structures. To our knowledge, this is the first work to systematically characterize the statistical gains from information borrowing in a non-asymptotic setting, providing both theoretical justification and practical criteria for selecting optimal prior depth in hierarchical modeling.

Technology Category

• Machine Learning: Bayesian Learning
• Reasoning under Uncertainty: Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this work, we offer a thorough analytical investigation into the role of shared hyperparameters in a hierarchical Bayesian model, examining their impact on information borrowing and posterior inference. Our approach is rooted in a non-asymptotic framework, where observations are drawn from a mixed-effects model, and a Gaussian distribution is assumed for the true effect generator. We consider a nested hierarchical prior distribution model to capture these effects and use the posterior means for Bayesian estimation. To quantify the effect of information borrowing, we propose an integrated risk measure relative to the true data-generating distribution. Our analysis reveals that the Bayes estimator for the model with a deeper hierarchy performs better, provided that the unknown random effects are correlated through a compound symmetric structure. Our work also identifies necessary and sufficient conditions for this model to outperform the one nested within it. We further obtain sufficient conditions when the correlation is perturbed. Our study suggests that the model with a deeper hierarchy tends to outperform the nested model unless the true data-generating distribution favors sufficiently independent groups. These findings have significant implications for Bayesian modeling, and we believe they will be of interest to researchers across a wide range of fields.

Problem

• Quantifying information borrowing in hierarchical Bayesian models
• Analyzing hyperparameter impact on posterior inference performance
• Establishing conditions for deeper hierarchy model superiority

Innovation

• Hierarchical Bayesian models with shared hyperparameters
• Non-asymptotic framework with Gaussian effect generator
• Compound symmetric correlation structure for better performance