A New Bound on the Cumulant Generating Function of Dirichlet Processes

Abstract

In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) $X\sim \text{DP}(\hat{I}\pm \hat{I}{½}_0)$, using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of $\hat{I}\pm \mapsto \log {\mathbb{E}}_{X\sim \text{DP}(\hat{I}\pm \hat{I}{½}_0)}[\exp ({\mathbb{E}}_X[\hat{I}\pm f])]$, where ${\mathbb{E}}_X[f]=\int fdX$. This result, combined with Fekete's lemma and Varadhan's integral lemma, converts the known asymptotic large deviation principle into a practical upper bound on the CGF $\log {\mathbb{E}}_{X\sim \text{DP}(\hat{I}\pm \hat{I}{½}_0)}\exp ({\mathbb{E}}_X[f])$ for any $\hat{I}\pm >0$. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence $\hat{I}\pm \mathrm{KL}(\hat{I}{½}_0\Vert \cdot )$. This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.