Approximate independence of permutation mixtures
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866915495779762176 |
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| author | Han, Yanjun Niles-Weed, Jonathan |
| author_facet | Han, Yanjun Niles-Weed, Jonathan |
| contents | We prove bounds on statistical distances between high-dimensional exchangeable mixture distributions (which we call \emph{permutation mixtures}) and their i.i.d. counterparts. Our results are based on a novel method for controlling $χ^2$ divergences between exchangeable mixtures, which is tighter than the existing methods of moments or cumulants. At a technical level, a key innovation in our proofs is a new Maclaurin-type inequality for elementary symmetric polynomials of variables that sum to zero and an upper bound on permanents of doubly-stochastic positive semidefinite matrices. We obtain as a corollary a new de Finetti-style theorem (in the language of Diaconis and Freedman, 1987), as well as several new statistical results, including a differential privacy guarantee for the ``shuffled privacy model'' with Gaussian noise and improved generic consistency guarantees for empirical Bayes procedures in compound decision problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_09341 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Approximate independence of permutation mixtures Han, Yanjun Niles-Weed, Jonathan Statistics Theory Information Theory Probability We prove bounds on statistical distances between high-dimensional exchangeable mixture distributions (which we call \emph{permutation mixtures}) and their i.i.d. counterparts. Our results are based on a novel method for controlling $χ^2$ divergences between exchangeable mixtures, which is tighter than the existing methods of moments or cumulants. At a technical level, a key innovation in our proofs is a new Maclaurin-type inequality for elementary symmetric polynomials of variables that sum to zero and an upper bound on permanents of doubly-stochastic positive semidefinite matrices. We obtain as a corollary a new de Finetti-style theorem (in the language of Diaconis and Freedman, 1987), as well as several new statistical results, including a differential privacy guarantee for the ``shuffled privacy model'' with Gaussian noise and improved generic consistency guarantees for empirical Bayes procedures in compound decision problems. |
| title | Approximate independence of permutation mixtures |
| topic | Statistics Theory Information Theory Probability |
| url | https://arxiv.org/abs/2408.09341 |