On the Minimum Attainable Risk in Permutation Invariant Problems

TL;DR

This work extends Robbins' compound decision framework to the broader class of permutation-invariant (PI) problems, including selective inference, by showing that the oracle PI rule is the Bayes rule under a uniform prior on all permutations of the fixed parameter vector $\boldsymbol{\theta}$. This yields an explicit greatest lower bound on the risk for PI procedures and reveals that empirical Bayes (EB) methods asymptotically attain this bound uniformly in $\boldsymbol{\theta}$ for many problems. The authors develop a unified decision-theoretic framework to accommodate PI structure, provide explicit oracle forms and computational strategies (e.g., permanents in the Poisson case), and illustrate applications to global null testing, multiple testing with FDR control, and selective inference. They prove an asymptotic attainability result for a PI problem involving the maximum observation, show how nuisance parameters can be incorporated, and discuss practical implications and future directions for extending PI theory beyond the independent setting. Overall, the paper broadens the scope of decision-theoretic guarantees from compound decisions to a wide range of PI problems, enabling principled EB-based procedures for selective and high-stakes inference.

Abstract

We consider a broad class of permutation invariant statistical problems by extending the standard decision theoretic definition to allow also selective inference tasks, where the target is specified only after seeing the data. For any such problem we show that, among all permutation invariant procedures, the minimizer of the risk at $\boldsymbolθ$ is precisely the rule that minimizes the Bayes risk under a (postulated) discrete prior assigning equal probability to every permutation of $\boldsymbolθ$. This gives an explicit characterization of the greatest lower bound on the risk of every sensible procedure in a wide range of problems. Furthermore, in a permutation invariant problem of estimating the parameter of a selected population under squared loss, we prove that this lower bound coincides asymptotically with a simpler lower bound, attained by the Bayes solution that replaces the aforementioned uniform prior on all permutations of $\boldsymbolθ$ by the i.i.d. prior with the same marginals. This has important algorithmic implications because it suggests that our greatest lower bound is asymptotically attainable uniformly in $\boldsymbolθ$ by an empirical Bayes procedure. Altogether, the above extends theory that has been established in the existing literature only for the very special case of compound decision problems.

Figures (1)

• Figure 1: Estimated ratio of the PI oracle risk to the simple oracle risk, Poisson example.

Theorems & Definitions (9)

• Definition 1: Permutation invariant decision problem
• Lemma 1
• proof
• Theorem 1
• proof
• Corollary 1
• Theorem 2
• Corollary 2
• proof : Proof of Theorem \ref{['thm:main']}