e-process (e-value)

Definition

An e-value $E$ is a nonnegative random variable with $E_P[E]\le 1$ ($\forall P\in H_0$) under the null hypothesis $H_0$. An e-process $(E_t)$ is a nonnegative process such that $E_\tau$ is an e-value at any stopping time $\tau$ ($E[E_\tau]\le1$) — typically a nonnegative supermartingale under the null. Then $1/E_t$ is an anytime-valid p-value, and Ville’s inequality $P\big(\exists t:\ E_t\ge 1/\alpha\big)\le \alpha$ gives time-uniform type-I error control → valid no matter when you stop or how much more you collect (optional stopping/continuation).

Intuitive Understanding

An “evidence accumulator” — a wealth process that grows by betting against the null (betting interpretation). Unlike a fixed-sample p-value, it stays valid no matter when you peek at it.

Related Concepts

• Confidence Sequence — inverting an e-process = time-uniform CI
• Off-Policy Evaluation (anytime-valid OPE) · Sensitivity Analysis

Key Papers

• Ramdas, Grünwald, Vovk & Shafer, “Game-Theoretic Statistics and Safe Anytime-Valid Inference”, Statistical Science 38(4):576–601, 2023
• Ramdas & Wang, “Hypothesis Testing with E-values”, FnT in Statistics, 2025 (arXiv:2410.23614) — the 390-page canonical reference