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| Main Authors: | , , , , , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.18621 |
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Table of Contents:
- In this paper, we introduce a novel approach for bounding the cumulant generating function (CGF) of a Dirichlet process (DP) $X \sim \text{DP}(αν_0)$, using superadditivity. In particular, our key technical contribution is the demonstration of the superadditivity of $α\mapsto \log \mathbb{E}_{X \sim \text{DP}(αν_0)}[\exp( \mathbb{E}_X[αf])]$, 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}(αν_0)}{\exp(\mathbb{E}_{X}{[f]})} $ for any $α> 0$. The bound is given by the convex conjugate of the scaled reversed Kullback-Leibler divergence $α\mathrm{KL}(ν_0\Vert \cdot)$. This new bound provides particularly effective confidence regions for sums of independent DPs, making it applicable across various fields.