On the Minimum Attainable Risk in Permutation Invariant Problems

We introduce a broad class of permutation invariant 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, the minimizer of the risk at $\boldsymbol{\theta}$ among all permutation invariant (equivariant) procedures is shown to be the Bayes rule that posits a uniform prior over all permutations of $\boldsymbol{\theta}$. This gives an explicit form of the greatest lower bound on the risk of any sensible procedure in a wide range of problems. From a practical perspective, approximations to the exact bound are required because of its computational cost. In a specific example of estimating the parameter of a selected population, we prove that our bound coincides asymptotically with the computationally tractable bound attained by the Bayes rule which replaces the uniform prior on all permutations of $\boldsymbol{\theta}$ by the i.i.d. prior with the same marginals. This generalizes results previously known only for the very special case of compound decision problems. The possibility of asymptotically attaining the latter bound by an empirical Bayes rule is discussed.