Shuffled bits in the low-detectability regime

We consider a decision problem in which data are unordered (unlabeled). Recent studies of this problem provide a complete asymptotic characterization of the decision performance for large data size, which is the solution of a convex optimization problem. While this is fully satisfactory from a numerical viewpoint, limited insight is offered because a closed-form explicit expression for the decision performance is, in general, not available. For binary observations and the challenging regime of low-detectability, we derive an extremely simple analytical solution, investigate its properties and discuss the obtained physical insights.