Approximate independence of permutation mixtures

2024-08-18

arXiv.org

AI Summary

This paper investigates statistical distance bounds between high-dimensional exchangeable mixture distributions (i.e., permutation mixtures) and their i.i.d. approximations. To overcome the challenge of controlling the χ²-divergence, we develop a novel analytical framework: (i) establishing Maclaurin-type inequalities for elementary symmetric polynomials of zero-mean variables; (ii) deriving tight upper bounds on the permanent of doubly stochastic positive semidefinite matrices; and (iii) integrating moment–cumulant methods, exchangeability structure, and asymptotic statistical theory. Key contributions include: (i) a strengthened de Finetti-type theorem with significantly improved convergence rates over classical versions; and (ii) a generalization of the Hannan–Robbins (1955) result on asymptotic optimality of compound decision rules to a broader class of exchangeable models, along with sharpened sufficient conditions and enhanced accuracy guarantees.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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).

Problem

• Bounds on statistical distances between exchangeable and i.i.d. mixtures
• Novel method for controlling χ² divergences in permutation mixtures
• New de Finetti-style theorem and statistical applications

Innovation

• Novel Maclaurin inequality for symmetric polynomials
• Upper bound on permanents of matrices
• Tighter control of chi-square divergences