Pre-Disclosure Experiment Menus: Oracle-Relative Risk and Joint Sample--Menu Asymptotics

Abstract

We study a resolution problem in local asymptotic decision theory: individual risks may admit Gaussian approximations that do not determine their vanishing difference. A finite menu of experiments is installed before context disclosure, although observations may be routed adaptively afterward. A greatest-element Blackwell order collapses adaptive routing to the best installed experiment and reduces the fixed-menu excess to an inverse-information distortion with frontier $A_k$. We develop differentiated, all-prior posterior transfer along a one-dimensional degradation chain and establish $F_{n,k_n}(H_n)=A_{k_n}\{1+o(1)\}$ for every diverging menu sequence with positive frontier and every admissible localization radius, without an additional direct sample-menu restriction. The transfer is exact under Gaussian degradation. Prior-free likelihood-generator conditions imply it for jump generators and are verified for binary attenuation, Poisson thinning, and negative-binomial thinning. If the distortion is uniformly quadratic on an Ahlfors-regular oracle image of dimension $r$, then $A_k\asymp k^{-2/r}$, and the original-scale excess mean squared error is of order $n^{-1}{k}^{-2/r}$. Calibrated Poisson sensor and radial-qubit measurement menus illustrate the result. A triangular Gaussian counterexample shows why pointwise Gaussian convergence is insufficient.

Main paper: 21 pages; supplementary material: 18 pages