Sharper Perturbed-Kullback-Leibler Exponential Tail Bounds for Beta and Dirichlet Distributions

TL;DR

This paper improves exponential tail bounds for Beta and Dirichlet distributions by introducing a larger perturbation in the KL divergence, resulting in tighter bounds and extending the results to Dirichlet processes.

Contribution

It introduces a novel method to tighten tail bounds for Beta and Dirichlet distributions by optimizing the perturbation in the KL divergence framework.

Findings

Tighter exponential tail bounds for Beta distributions.

Extension of bounds to Dirichlet distributions and processes.

Enhanced understanding of distribution tail behaviors.

Abstract

This paper presents an improved exponential tail bound for Beta distributions, refining a result in [15]. This improvement is achieved by interpreting their bound as a regular Kullback-Leibler (KL) divergence one, while introducing a specific perturbation $\eta$ that shifts the mean of the Beta distribution closer to zero within the KL bound. Our contribution is to show that a larger perturbation can be chosen, thereby tightening the bound. We then extend this result from the Beta distribution to Dirichlet distributions and Dirichlet processes (DPs).