Aggregation of Statistical Evidence under Exchangeability
Machine Learning
Summary
The authors present a method to combine statistical evidence when the relationships between data points are unknown or complicated, using a concept called group-invariance. They base their approach on transforming data sets and treating these transformed sets equally to better understand the evidence. Their method improves upon traditional ways of controlling error rates, adapts to unknown dependencies, and can be used in both fixed and sequential testing setups. They also provide practical versions that reduce computational effort and show applications in advanced testing and prediction methods.
Authors
Antonin Schrab, Rajen Shah, Arthur Gretton, Ilmun Kim
Abstract
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.