Approximate independence of permutation mixtures (2408.09341v2)

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).

• The paper introduces a novel χ² divergence control method that ensures bounded divergence in high-dimensional permutation mixtures.
• The paper derives a De Finetti-style theorem, reinforcing compound decision theory with improved approximations for exchangeable sequences.
• The paper applies its methodology to Gaussian and Bernoulli models, demonstrating practical benefits for empirical Bayes and high-dimensional inference.

Approximate Independence of Permutation Mixtures

The paper "Approximate independence of permutation mixtures" by Yanjun Han and Jonathan Niles-Weed focuses on the analysis of statistical properties of high-dimensional exchangeable distributions, termed permutation mixtures, and compares them to their independent, identically distributed (i.i.d.) analogues. The authors introduce and utilize a novel approach to control the $\chi^2$ divergence between permutation mixtures, achieving results that are tighter than those obtained through traditional methods based on moments or cumulants.

Main Contributions

The paper introduces several key contributions:

1. Novel $\chi^2$ Divergence Control: The authors develop a new method for managing $\chi^2$ divergences between exchangeable mixtures, leveraging inequalities for elementary symmetric polynomials and upper bounds on the permanents of certain matrices.
2. De Finetti-style Theorem and Compound Decision Problems: By applying their $\chi^2$ divergence results, the authors derive a De Finetti-style theorem. They also generalize and reinforce the results surrounding compound decision theory established by Hannan and Robbins.
3. Permutations in Practical Models: The paper covers a broad range of practical scenarios where permutation mixtures are relevant, such as compound decisions, empirical Bayes methods, De Finetti theorems for exchangeable sequences, mean-field approximations, and statistical distances within information geometry.

Technical Innovations

Permutation Mixtures and $\chi^2$ Divergence

Permutation mixtures are defined by a collection of probability measures $\{P_i\}_{i=1}^{n}$ on a shared probability space. The novel aspect of the authors’ approach lies in their control of the $\chi^2$ divergence between permutation mixtures and their i.i.d. counterparts. This comparison is crucial because traditional bounds often fail to capture the subtle statistical differences in high-dimensional contexts.

Key Mathematical Tools

1. Maclaurin-Type Inequality: A newly established inequality for elementary symmetric polynomials of variables that sum to zero.
2. Matrix Permanent Bounds: Based on the permanents of doubly-stochastic positive semidefinite matrices, which enable the development of tighter $\chi^2$ divergence upper bounds.

Results and Findings

1. Asymptotic Behavior of $\chi^2$ Divergence: The authors prove that in high-dimensional settings, $\chi^2(\mathbf{P}_n \| \mathbf{Q}_n)$ remains bounded independent of the dimension $n$, establishing asymptotic properties that allow analysis of $\mathbf{P}_n$ under simpler product measures $\mathbf{Q}_n$.
2. De Finetti-style Theorem: The results imply new approximations for general exchangeable sequences, reinforcing and extending classical ideas in statistical theory.
3. General Models: The paper shows the practical applications of permutation mixtures to Gaussian and Bernoulli families, providing specialized results such as scaling behaviors for $\chi^2$ divergence in these contexts.

Implications and Future Directions

This paper has significant implications for both theoretical and practical aspects of statistics and information theory.

• Empirical Bayes and Compound Decisions: The results give rise to new risk bounds within empirical Bayes frameworks, suggesting more effective strategies for statistical decision making in compound settings.
• Improved Statistical Distances: The rigorous bounds on $\chi^2$ divergence imply that permutation mixtures can be effectively approximated by simpler i.i.d. models without sacrificing accuracy. This understanding potentially simplifies the analysis and computation in high-dimensional statistics.
• Potential in High-dimensional Inference: The techniques could be extended to more complex mixture models, fostering advancements in areas such as high-dimensional inference and machine learning.

Conclusion

Han and Niles-Weed have made a substantial contribution by introducing a robust framework for analyzing permutation mixtures using $\chi^2$ divergence. Their work expands the theoretical foundation of high-dimensional exchangeable distributions and paves the way for further exploration in both high-dimensional statistics and practical applications in empirical Bayes and decision theory.