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| Main Authors: | , , , , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.18621 |
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| _version_ | 1866914958586937344 |
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| author | Perrault, Pierre Belomestny, Denis Ménard, Pierre Moulines, Éric Naumov, Alexey Tiapkin, Daniil Valko, Michal |
| author_facet | Perrault, Pierre Belomestny, Denis Ménard, Pierre Moulines, Éric Naumov, Alexey Tiapkin, Daniil Valko, Michal |
| 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_18621 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A New Bound on the Cumulant Generating Function of Dirichlet Processes Perrault, Pierre Belomestny, Denis Ménard, Pierre Moulines, Éric Naumov, Alexey Tiapkin, Daniil Valko, Michal Probability Information Theory 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. |
| title | A New Bound on the Cumulant Generating Function of Dirichlet Processes |
| topic | Probability Information Theory |
| url | https://arxiv.org/abs/2409.18621 |