Statistical reduction before the target is known: two boundary results
Abstract
Suppose that the eventual use of data is not known when the data are reduced or collected. This note considers two simple boundary cases. In a finite statistical experiment, a statistic preserves the Bayes risk for every finite later decision problem if and only if it is sufficient. Hence, when the minimal sufficient statistic is one-to-one, exact preservation of all later decision problems permits no nontrivial reduction. We then consider adaptive sampling from m independent Gaussian streams when an external query specifies the coordinate to be classified only after sampling stops. Under coordinatewise error control, the optimal symmetric average sample size is exactly m times the one-coordinate optimum. A change-of-measure argument gives the corresponding pointwise lower bound in terms of binary relative entropy.
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BibTeX
@misc{heide2026statistical,
title = {{Statistical reduction before the target is known: two boundary results}},
author = {Rianne de Heide},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05286},
eprint = {2609.05286},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05286},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Statistical reduction before the target is known: two boundary results
AU - Rianne de Heide
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05286
UR - https://arxiv.org/abs/2609.05286
AB - Suppose that the eventual use of data is not known when the data are reduced or collected. This note considers two simple boundary cases. In a finite statistical experiment, a statistic preserves the Bayes risk for every finite later decision problem if and only if it is sufficient. Hence, when the minimal sufficient statistic is one-to-one, exact preservation of all later decision problems permits no nontrivial reduction. We then consider adaptive sampling from m independent Gaussian streams when an external query specifies the coordinate to be classified only after sampling stops. Under coordinatewise error control, the optimal symmetric average sample size is exactly m times the one-coordinate optimum. A change-of-measure argument gives the corresponding pointwise lower bound in terms of binary relative entropy.
ER -