P-values are probabilities. Likelihood ratios are ratios. Bayes factors are weighted averages. E-values are wealth.
The paper (arXiv:2603.24421, March 2026) frames statistical evidence as a betting game. A bettor wagers against the null hypothesis. At each observation, the bettor places a stake — more where the data looks incompatible with the null, less where it looks compatible. The e-value at any point is the bettor's total wealth, starting from 1.
This reframing is not a metaphor. It is a mathematical identity. An e-value is defined as any nonnegative random variable with expectation at most 1 under the null hypothesis — exactly the condition that prevents the bettor from making money on average when the null is true. A sequence of observations generates an e-process: a running wealth total that can be evaluated at any stopping time without adjustment.
The key advantage is optional stopping. P-values require that you fix the sample size in advance; peeking at the data inflates the false positive rate. Likelihood ratios have their own calibration issues. But e-values are valid at any stopping time, including data-dependent ones. You can stop when the evidence is strong enough, continue when it's ambiguous, and combine evidence across studies by multiplication. The wealth interpretation makes this natural: stopping when you're rich enough to be convinced is not a statistical sin — it is rational.
The paper argues that e-values combine the best features of existing evidence measures: the objectivity of p-values (no prior required for simple nulls), the interpretability of likelihood ratios (how much the data favors one hypothesis over another), and the composability of Bayes factors (natural combination across experiments). The price is power — e-values are less efficient than p-values in fixed-sample designs. But the gain is flexibility, and in sequential and optional-stopping settings, the gain exceeds the cost.