🤖 AI Summary
Existing exponential tail bounds for Beta and Dirichlet distributions—particularly KL-divergence-based bounds—are loose, especially in high-dimensional or nonparametric Bayesian settings.
Method: We propose a novel KL-divergence perturbation framework that introduces a mean-zero shift to tighten these bounds systematically. This perturbation is optimized to enhance bound sharpness while preserving statistical validity.
Contribution/Results: Our approach is the first to extend perturbation-based optimization from univariate Beta to multivariate Dirichlet distributions and, further, to the nonparametric Dirichlet process. Theoretically, increasing the perturbation parameter strictly improves bound tightness; the resulting bounds dominate classical Chernoff-type and standard KL-type bounds across diverse parameter regimes. Empirical evaluation demonstrates superior performance in constructing confidence intervals and analyzing posterior contraction rates in high-dimensional probabilistic inference. By bridging exponential inequalities with Bayesian nonparametrics, our work expands the theoretical applicability of concentration bounds in modern Bayesian statistics.
📝 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 $η$ 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).