Resolution Is Not Accessibility: Coupled Minimax Risk for Reduced Statistical Experiments

Abstract

A reduced experiment can have vanishing minimax risk yet lose arbitrarily much precision relative to full observation. We determine when growing reductions preserve target-specific minimax precision, their exact cost, and which omissions can remain without increasing that cost. For Gaussian sequence reductions, we give necessary and sufficient criteria for bounded relative risk and losslessness. With parameter-dependent information, sampling difficulty and unresolved target variation draw on the same parameter budget. We prove that the exact global asymptotic minimax risk is the maximum scalar interval-decision risk over their jointly attainable frontier under a Sobolev constraint. This identity shows that positive ambiguity at the inverse-square-root sample-size scale can preserve the full-observation minimax constant. Under the corresponding full-experiment risk match, a sharp boundary criterion determines when critical-order resolution suffices for losslessness. A quadratic Poisson budget yields an exact threshold and quadratic risk onset under a stated information condition. Across Poisson power budgets, separate-maxima planning incurs a diverging asymptotic coordinate-budget factor near full precision, both with a positive lossless threshold and with zero threshold but higher-order risk onset. Finite-sample calculations and conditional medical-data and classifier audits distinguish the effects of observation coverage and repeated measurement.