Measuring Evidence against Exchangeability and Group Invariance with E-values

TL;DR

The paper develops a comprehensive framework for measuring evidence against exchangeability and general group invariance using e-values and e-processes. It establishes that valid e-values for group invariance can be constructed via orbit-wise averaging, with Monte Carlo implementations, and characterizes when such e-values are exact and optimal. By formulating orbit-decomposed utilities, it derives both log-optimal and Neyman–Pearson-type e-values, and extends to sequential settings through orbit-wise martingales and test martingales. The work further connects to ergodic theory for non-compact groups and provides practical illustrations, including hot-hand analysis and Gaussian sphericity, with simulations showing substantial gains over traditional tests. These contributions enable flexible, anytime-valid inference for invariance properties in a broad range of statistical settings, with concrete guidance on construction, computation, and interpretation of evidence against invariance.

Abstract

We study e-values for quantifying evidence against exchangeability and general invariance of a random variable under a compact group. We start by characterizing such e-values, and explaining how they nest traditional group invariance tests as a special case. We show they can be easily designed for an arbitrary test statistic, and computed through Monte Carlo sampling. We prove a result that characterizes optimal e-values for group invariance against optimality targets that satisfy a mild orbit-wise decomposition property. We apply this to design expected-utility-optimal e-values for group invariance, which include both Neyman-Pearson-optimal tests and log-optimal e-values. Moreover, we generalize the notion of rank- and sign-based testing to compact groups, by using a representative inversion kernel. In addition, we characterize e-processes for group invariance for arbitrary filtrations, and provide tools to construct them. We also describe test martingales under a natural filtration, which are simpler to construct. Peeking beyond compact groups, we encounter e-values and e-processes based on ergodic theorems. These nest e-processes based on de Finetti's theorem for testing exchangeability.

Figures (2)

• Figure 1: Plots of 1 000 e-processes over the number of arrived batches. The highlighted lines are running quantiles: x% of the e-processes have not crossed above the line at the indicated time. The plot on the left is under the null hypothesis, and the plot on the right is under the alternative. The horizontal dotted line is at 20.
• Figure 2: Plots of 1 000 e-processes over the number of arrived observations under a normal alternative with mean $m = 1$. The highlighted lines are running quantiles: x% of the e-processes have not crossed above the line at the indicated time. The plot on the left is for our log-optimal e-value-based e-process, and the plot on the right is for the one based on de1999general. The horizontal dotted line is at 20.

Theorems & Definitions (78)

• Example 1: Orthonormal matrices
• Example \ref{exm:rotation} (Part B)
• Example 2: One orbit
• Definition 1: Invariant probability
• Example \ref{exm:rotation} (Part C)
• Lemma 1: Equivalent definitions of invariance
• Example \ref{exm:rotation} (Part C)
• Definition 2: Invariance through a statistic
• Remark 1
• Remark 2: Inversion kernels, ranks, signs and normalization