Convergence of Statistical Estimators via Mutual Information Bounds

Summary

The paper introduces a novel mutual information bound for statistical models, which yields improved contraction rates for fractional posteriors in Bayesian nonparametrics and can be used to study various estimation methods, including variational inference and Maximum Likelihood Estimation (MLE).

Highlights

• The bound has wide-ranging applications in statistical inference.
• It improves contraction rates for fractional posteriors in Bayesian nonparametrics.
• It can be used to study variational inference and Maximum Likelihood Estimation (MLE).
• The bound is derived using a localization technique of Catoni.
• It eliminates suboptimal logarithmic terms in existing bounds.
• The paper also discusses the application of the bound to the MLE.
• The results rely on two sets of assumptions: Assumptions 2, 3, and 4 on model complexity, and an additional Assumption 1 on the model.

Key Insights

• The mutual information bound provides a new tool to study a wide range of statistical methods, including Bayesian and non-Bayesian approaches.
• The bound is particularly useful in Bayesian nonparametrics, where it leads to improved contraction rates for fractional posteriors.
• The localization technique of Catoni is crucial in deriving the bound, as it allows for a more precise control of the prior distribution.
• The elimination of suboptimal logarithmic terms in existing bounds is a significant improvement, as it leads to tighter convergence rates.
• The application of the bound to the MLE is an important contribution, as it demonstrates the versatility of the approach.
• The reliance on two sets of assumptions highlights the need for careful consideration of model complexity and regularity conditions in statistical analysis.
• The paper's results have implications for the study of statistical inference, as they provide new insights into the behavior of various estimation methods.

Mindmap

Citation

Khribch, E. M., & Alquier, P. (2024). Convergence of Statistical Estimators via Mutual Information Bounds (Version 1). arXiv. https://doi.org/10.48550/ARXIV.2412.18539