在复杂依赖下,用置换构造可交换数据集,实现更灵敏的统计证据聚合。
Aggregation of Statistical Evidence under Exchangeability
- 基于置换生成可交换数据集,通过变换后证据聚合与校准。
- 有限样本下优于传统方法,在任意依赖下仍保持有效性且自适应依赖结构。
- 支持顺序和数据依赖聚合,适合自适应非参数检验与预测校准场景。
我们在未知且可能复杂的依赖结构下研究统计证据的聚合问题,采用群不变性思想。基于置换构造的可交换数据集,将每个变换后的数据集对应的统计量进行证据聚合,并在变换间对聚合结果进行校准。本文建立了该框架的有限样本功效与自适应性理论,扩展至顺序和数据依赖的聚合方式,均保持有效性。对于单批聚合(即同一组变换同时用于标准化与校准),我们证明其临界值在任意依赖下均优于确定性校准方法(如Bonferroni校正),并能自适应未知依赖结构。此外,提出一种顺序alpha支出版本,可在强证据时提前拒绝;还引入双批扩展,分离标准化与校准过程,以支持学习型聚合规则并降低计算成本。在自适应非参数检验与置信推断中的应用表明,该方法显著提升了现有聚合方法的性能。
原文摘要 · Abstract (English)
We study aggregation of statistical evidence under unknown and potentially complex dependence using group-invariance. Building on permutation-based constructions that treat transformed datasets as exchangeable units, we aggregate evidence across statistics for each transformed dataset and calibrate the resulting aggregates across transformations. We develop a finite-sample power and adaptivity theory for this framework, together with extensions to sequential and data-dependent aggregation that preserve validity. For single-batch aggregation, which uses one collection of transformed datasets for both standardization and calibration, we show that the critical values uniformly improve on deterministic calibrations valid under arbitrary dependence, including Bonferroni correction, while adapting to the unknown dependence structure. We also introduce a sequential alpha-spending version that permits early rejection when evidence is strong, and a two-batch extension that separates standardization from calibration to accommodate learned aggregation rules and reduce computation. Applications to adaptive nonparametric testing and conformal prediction illustrate how these results sharpen existing aggregation methods.
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