Convergence of Statistical Estimators via Mutual Information Bounds

TL;DR

This paper introduces a new mutual information bound that enhances understanding of statistical inference limits and improves contraction rates in Bayesian nonparametrics, connecting information theory with various estimation methods.

Contribution

It presents a novel mutual information bound applicable across multiple statistical inference techniques, advancing theoretical understanding and practical analysis.

Findings

Improved contraction rates for fractional posteriors in Bayesian nonparametrics.

A new mutual information bound applicable to diverse estimation methods.

Enhanced theoretical connections between information theory and statistical inference.

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.