用互信息新界改进统计估计的收敛性分析。
Convergence of Statistical Estimators via Mutual Information Bounds
- 提出统计模型的新型互信息上界，统一分析框架。
- 在贝叶斯非参数中提升分数后验的收缩速率。
- 适用于变分推断、最大似然等广泛估计方法。
统计学习理论近期揭示了互信息（MI）界、PAC-贝叶斯理论与贝叶斯非参数之间的深刻联系。本文提出一种针对统计模型的新型互信息界，具有广泛的应用前景。该界可改进贝叶斯非参数中分数后验的收缩速率，亦可用于研究变分推断、最大似然估计（MLE）等多种估计方法。通过连接这些不同领域，本工作深化了对统计推断基本极限及信息在数据学习中作用的理解。我们希望这些结果不仅能厘清统计推断与信息论的关联，还能为研究各类估计器提供新的工具箱。
原文摘要 · Abstract (English)
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.
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