# Hypothesis testing with e-values

arXiv: https://arxiv.org/abs/2410.23614 · HTML: https://mlsanity.com/paper/2410.23614/

## Key facts

- Cited by 74 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2024.

## Built on by (most cited first, 30 of 74)

- [E-Values Expand the Scope of Conformal Prediction](https://mlsanity.com/paper/2503.13050/) (2025)
- [E-valuator: Reliable Agent Verifiers with Sequential Hypothesis Testing](https://mlsanity.com/paper/2512.03109/) (2025)
- [Prediction-Powered E-Values](https://mlsanity.com/paper/2502.04294/) (2025)
- [Backward Conformal Prediction](https://mlsanity.com/paper/2505.13732/) (2025)
- [Conformal Calibration: Ensuring the Reliability of Black-Box AI in Wireless Systems](https://mlsanity.com/paper/2504.09310/) (2025)
- [Certified Self-Consistency: Statistical Guarantees and Test-Time Training for Reliable Reasoning in LLMs](https://mlsanity.com/paper/2510.17472/) (2025)
- [Conformal Selective Prediction with General Risk Control](https://mlsanity.com/paper/2603.24704/) (2026)
- [Adaptive Prediction-Powered AutoEval with Reliability and Efficiency Guarantees](https://mlsanity.com/paper/2505.18659/) (2025)
- [On Continuous Monitoring of Risk Violations under Unknown Shift](https://mlsanity.com/paper/2506.16416/) (2025)
- [On Stopping Times of Power-one Sequential Tests: Tight Lower and Upper Bounds](https://mlsanity.com/paper/2504.19952/) (2025)
- [Anytime-valid, Bayes-assisted, Prediction-Powered Inference](https://mlsanity.com/paper/2505.18000/) (2025)
- [Sequentially Auditing Differential Privacy](https://mlsanity.com/paper/2509.07055/) (2025)
- [Adaptive Coverage Policies in Conformal Prediction](https://mlsanity.com/paper/2510.04318/) (2025)
- [Theoretical guarantees for change localization using conformal p-values](https://mlsanity.com/paper/2510.08749/) (2025)
- [Optimistic Interior Point Methods for Sequential Hypothesis Testing by Betting](https://mlsanity.com/paper/2502.07774/) (2025)
- [A Collectivist, Economic Perspective on AI](https://mlsanity.com/paper/2507.06268/) (2025)
- [Auditing Pay-Per-Token in Large Language Models](https://mlsanity.com/paper/2510.05181/) (2025)
- [Stopping Rules for Stochastic Gradient Descent via Anytime-Valid Confidence Sequences](https://mlsanity.com/paper/2512.13123/) (2025)
- [Asymptotically optimal sequential change detection for bounded means](https://mlsanity.com/paper/2602.05272/) (2026)
- [Behavioral Fingerprints for LLM Endpoint Stability and Identity](https://mlsanity.com/paper/2603.19022/) (2026)
- [Efficient Sequential Evaluation of Large Language Models](https://mlsanity.com/paper/2607.17409/) (2026)
- [Ensuring Reliability via Hyperparameter Selection: Review and Advances](https://mlsanity.com/paper/2502.04206/) (2025)
- [Toward Universal Laws of Outlier Propagation](https://mlsanity.com/paper/2502.08593/) (2025)
- [Universality of conformal prediction under the assumption of randomness](https://mlsanity.com/paper/2502.19254/) (2025)
- [Prediction-Powered Risk Monitoring of Deployed Models for Detecting Harmful Distribution Shifts](https://mlsanity.com/paper/2602.02229/) (2026)
- [Adaptive auditing of AI systems with anytime-valid guarantees](https://mlsanity.com/paper/2605.07002/) (2026)
- [Extending Prediction-Powered Inference through Conformal Prediction](https://mlsanity.com/paper/2510.16166/) (2025)
- [E-Scores for (In)Correctness Assessment of Generative Model Outputs](https://mlsanity.com/paper/2510.25770/) (2025)
- [Eventually LIL Regret: Almost Sure lnln T Regret for a sub-Gaussian Mixture on Unbounded Data](https://mlsanity.com/paper/2512.12325/) (2025)
- [Online LLM watermark detection via e-processes](https://mlsanity.com/paper/2602.14286/) (2026)

## Abstract (arXiv)

This book is written to offer a humble, but unified, treatment of e-values in hypothesis testing. It is organized into three parts: Fundamental Concepts, Core Ideas, and Advanced Topics. The first part includes four chapters that introduce the basic concepts. The second part includes five chapters of core ideas such as universal inference, log-optimality, e-processes, operations on e-values, and e-values in multiple testing. The third part contains seven chapters of advanced topics. The book collates important results from a variety of modern papers on e-values and related concepts, and also contains many results not published elsewhere. It offers a coherent and comprehensive picture on a fast-growing research area, and is ready to use as the basis of a graduate course in statistics and related fields.