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

arXiv: https://arxiv.org/abs/2508.07991 · HTML: https://mlsanity.com/paper/2508.07991/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 26 references that are in the map.
- First posted on arXiv 11 Aug 2025.

## Builds on (most cited first, 5 of 5)

- [From Dirichlet to Rubin: Optimistic Exploration in RL without Bonuses](https://mlsanity.com/paper/2205.07704/) (2022)
- [KL-UCB-switch: optimal regret bounds for stochastic bandits from both a distribution-dependent and a distribution-free viewpoints](https://mlsanity.com/paper/1805.05071/) (2018)
- [On the Non-asymptotic and Sharp Lower Tail Bounds of Random Variables](https://mlsanity.com/paper/1810.09006/) (2018)
- [Bayesian Adaptive Data Analysis Guarantees from Subgaussianity](https://mlsanity.com/paper/1611.00065/) (2016)
- [Sharp Deviations Bounds for Dirichlet Weighted Sums with Application to analysis of Bayesian algorithms](https://mlsanity.com/paper/2304.03056/) (2023)

## Abstract (arXiv)

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 η 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).