Approximate independence of permutation mixtures
Abstract
We prove bounds on statistical distances between high-dimensional exchangeable mixture distributions (which we call permutation mixtures) and their i.i.d. counterparts. Our results are based on a novel method for controlling $\chi^2$ 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. Our results imply a de Finetti-style theorem (in the language of Diaconis and Freedman, 1987) and general asymptotic results for compound decision problems, generalizing and strengthening a result of Hannan and Robbins (1955).
- Publication:
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arXiv e-prints
- Pub Date:
- August 2024
- DOI:
- 10.48550/arXiv.2408.09341
- arXiv:
- arXiv:2408.09341
- Bibcode:
- 2024arXiv240809341H
- Keywords:
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- Mathematics - Statistics Theory;
- Computer Science - Information Theory;
- Mathematics - Probability
- E-Print:
- Added the reference Tang and Polyanskiy (2023) and fixed some typos