Convergence of Statistical Estimators via Mutual Information Bounds

2024-12-24

AI Summary

This work addresses the precise characterization of information-theoretic limits and convergence rates in Bayesian nonparametrics. Focusing on fractional posteriors, variational inference, and maximum likelihood estimation, we derive a universal upper bound on mutual information that uniformly governs their statistical convergence rates. Our bound substantially improves existing contraction rate guarantees for fractional posteriors and, for the first time, establishes a rigorous theoretical bridge between the PAC-Bayes framework and Bayesian nonparametric information theory. The results yield tighter, quantifiable convergence rates and unify disparate inferential paradigms under a single information-theoretic analytical framework. This provides a principled, comparable benchmark for fundamental statistical limits in nonparametric settings.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Relational Probabilistic Models
• Search and Optimization: Non-convex Optimization

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: Psychology-informed user models and recommender systems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Recent advances in statistical learning theory have revealed profound connections between mutual information (MI) bounds, PAC-Bayesian theory, and Bayesian nonparametrics. This work introduces a novel mutual information bound for statistical models. The derived bound has wide-ranging applications in statistical inference. It yields improved contraction rates for fractional posteriors in Bayesian nonparametrics. It can also be used to study a wide range of estimation methods, such as variational inference or Maximum Likelihood Estimation (MLE). By bridging these diverse areas, this work advances our understanding of the fundamental limits of statistical inference and the role of information in learning from data. We hope that these results will not only clarify connections between statistical inference and information theory but also help to develop a new toolbox to study a wide range of estimators.

Problem

• Statistical Learning Theory
• Bayesian Nonparametric Statistics
• Information Boundaries

Innovation

• Mutual Information Bound
• Bayesian Nonparametric Statistics
• Data Estimation Methods