Approximate independence of permutation mixtures

TL;DR

This paper establishes bounds on the statistical distances between permutation mixture distributions and their i.i.d. equivalents, introducing novel inequalities and applying them to derive new theoretical results in statistics and privacy.

Contribution

It introduces a new method for controlling $ ext{chi}^2$ divergences in exchangeable mixtures, leading to a new de Finetti-style theorem and several statistical guarantees.

Findings

Bounds on statistical distances between permutation mixtures and i.i.d. distributions

A new de Finetti-style theorem for exchangeable mixtures

Differential privacy guarantees in the shuffled privacy model

Abstract

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.