Approximate independence of permutation mixtures

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 $\chi^2$ 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. 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.