A New Bound on the Cumulant Generating Function of Dirichlet Processes

TL;DR

This paper derives a new upper bound on the cumulant generating function of Dirichlet processes using superadditivity, enabling better confidence regions for sums of independent DPs.

Contribution

It introduces a novel superadditivity-based approach to bound the CGF of Dirichlet processes, connecting large deviation principles with practical upper bounds.

Findings

Provides a convex conjugate bound involving KL divergence.

Enables construction of effective confidence regions for sums of DPs.

Bridges asymptotic theory with practical statistical bounds.

Abstract

In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) $X \sim \text{DP}(\alpha \nu_0)$, using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of $\alpha \mapsto \log \mathbb{E}_{X \sim \text{DP}(\alpha \nu_0)}[\exp( \mathbb{E}_X[\alpha f])]$, where $\mathbb{E}_X[f] = \int f dX$. 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}(\alpha\nu_0)}{\exp(\mathbb{E}_{X}{[f]})} $ for any $\alpha > 0$. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence $\alpha\mathrm{KL}(\nu_0\Vert \cdot)$. This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.