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).

翻译：我们证明了高维可交换混合分布（我们称之为置换混合分布）与其独立同分布对应分布之间统计距离的界。我们的结果基于一种控制可交换混合分布间$\chi^2$散度的新方法，该方法比现有的矩方法或累积量方法更为严格。在技术层面，我们证明中的关键创新包括：针对求和为零的变量的初等对称多项式提出新的麦克劳林型不等式，以及对双随机半正定矩阵的积和式给出上界。我们的结果蕴含了德菲内蒂风格定理（采用Diaconis和Freedman于1987年提出的表述方式），并为复合决策问题提供了广义渐近结果，从而推广并强化了Hannan和Robbins（1955）的结论。