  Simultaneous statistical inference problems are at the basis of almost any
scientific discovery process. We consider a class of simultaneous inference
problems that are invariant under permutations, meaning that all components of
the problem are oblivious to the labelling of the multiple instances under
consideration. For any such problem we identify the optimal solution which is
itself permutation invariant, the most natural condition one could impose on
the set of candidate solutions. Interpreted differently, for any possible value
of the parameter we find a tight (non-asymptotic) lower bound on the
statistical performance of any procedure that obeys the aforementioned
condition. By generalizing the standard decision theoretic notions of
permutation invariance, we show that the results apply to a myriad of popular
problems in simultaneous inference, so that the ultimate benchmark for each of
these problems is identified. The connection to the nonparametric empirical
Bayes approach of Robbins is discussed in the context of asymptotic
attainability of the bound uniformly in the parameter value.
