Measuring Evidence against Exchangeability and Group Invariance with E-values
Proposes e-values for quantifying evidence against exchangeability and invariance, with Monte Carlo computation and optimal design.
Key Findings
Methodology
This paper introduces e-values as tools to quantify evidence against the invariance of a random variable under group actions. It begins by defining e-values that satisfy specific expectation conditions, enabling efficient Monte Carlo estimation. Next, it derives conditions for optimal e-values under orbit-wise decomposition, maximizing expected utility with respect to various utility functions, including Neyman-Pearson and log utility. The approach extends rank- and sign-based tests to compact groups via a representative inversion kernel. Furthermore, it constructs e-processes and test martingales for arbitrary filtrations, generalizing to non-compact groups using ergodic theorems based on de Finetti’s theorem, facilitating sequential invariance testing.
Key Results
- The proposed e-values satisfy the condition EG[Gx]≤1, allowing Monte Carlo approximation. Empirical results on simulated data show that these e-values outperform traditional p-values in detecting violations of invariance, with higher power and robustness.
- Optimal e-values under orbit-wise decomposition achieve maximal expected utility, especially for log and Neyman-Pearson objectives, demonstrating superior performance across sample sizes and complex group structures.
- Extending to non-compact groups, the use of ergodic theorems enables the construction of e-processes for sequential testing of exchangeability, adaptable to streaming data environments.
Significance
This work advances the theory of hypothesis testing by integrating e-values into group invariance testing, providing a flexible, continuous, and anytime-valid measure of evidence. It addresses limitations of classical binary tests, offering a unified framework that combines optimality, computational efficiency, and applicability to sequential and high-dimensional data. The methods have broad implications for statistical modeling, machine learning, causal inference, and multiple testing, enabling more robust and adaptive inference in complex settings.
Technical Contribution
The paper systematically derives the form of e-values for group invariance, establishes conditions for optimality under orbit-wise criteria, and develops Monte Carlo algorithms for efficient computation. It introduces a novel extension of e-processes via ergodic theorems, constructs test martingales under natural filtrations, and generalizes the framework to non-compact groups, enriching the theoretical toolkit for sequential and invariant hypothesis testing.
Novelty
This is the first comprehensive framework integrating e-values into group invariance testing, deriving explicit formulas for optimal e-values, and extending the theory to non-compact groups via ergodic theorems. It bridges the gap between classical permutation tests and modern sequential inference, offering a continuous evidence measure that is both theoretically optimal and practically scalable.
Limitations
- The approach relies on the orbit decomposition property, which may not hold for certain complex or non-smooth group actions, limiting applicability.
- Monte Carlo sampling, while efficient, can become computationally intensive in high-dimensional or highly complex group settings.
- Extension to non-ergodic or non-invariant models remains challenging, requiring further theoretical development.
Future Work
Future research could explore adaptive sampling strategies, integration with deep learning models for invariance learning, and extension to broader classes of non-compact or infinite-dimensional groups. Developing scalable algorithms for high-dimensional data and applying these methods to real-world problems like causal inference and fairness testing are promising directions.
AI Executive Summary
This paper introduces a novel framework for measuring evidence against the invariance of a random variable under group actions using e-values. Traditional invariance tests, such as permutation tests, are binary and often limited to specific groups or assumptions. In contrast, the authors propose a flexible, continuous measure of evidence that can be computed efficiently via Monte Carlo sampling. They establish a characterization of valid e-values for group invariance, showing that EG[Gx]≤1 for a uniform group element G, and derive explicit formulas involving test statistics. The framework encompasses classical Neyman-Pearson optimal tests and extends to log-optimal e-values, providing a unified approach to invariance testing. The authors further develop methods for constructing e-processes and test martingales applicable to arbitrary filtrations, including non-compact groups via ergodic theorems. Empirical experiments demonstrate that these e-values outperform traditional p-values in detecting violations of invariance, especially in high-dimensional and complex group structures. The approach offers significant advantages for sequential analysis, multiple testing, and model validation in modern data science. Overall, this work bridges classical hypothesis testing and modern sequential inference, delivering a powerful, scalable, and theoretically grounded toolkit for invariance testing across diverse applications. Future directions include integrating deep learning for invariance discovery, improving computational scalability, and extending to broader group classes, promising impactful advances in statistical methodology and data analysis.
Deep Dive
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.