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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 \u201cshuffled privacy model\u201d with Gaussian noise and improved generic consistency guarantees for empirical Bayes procedures in compound decision problems.<\/jats:p>","DOI":"10.1214\/26-aos2619","type":"journal-article","created":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T13:53:31Z","timestamp":1782395611000},"page":"1594-1621","source":"Crossref","is-referenced-by-count":2,"title":["Approximate independence of permutation mixtures"],"prefix":"10.1214","volume":"54","author":[{"given":"Yanjun","family":"Han","sequence":"first","affiliation":[{"name":"Courant Institute of Mathematical Sciences and Center for Data Science, New York University"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jonathan","family":"Niles-Weed","sequence":"additional","affiliation":[{"name":"Courant Institute of Mathematical Sciences and Center for Data Science, New York University"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"108","reference":[{"key":"16","doi-asserted-by":"publisher","unstructured":"<small>Diaconis, P.<\/small> (1977). 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