e-process

An e-process is the sequential analogue of the e-value. They are one of the foundational tools in safe, anytime-valid inference (SAVI).

We say a stochastic process $(E_t)$ is an e-process for $\mathcal{P}$ if it is an e-value at every stopping-time:

$$\text{for all stopping times }\tau ,\underset{P\in \mathcal{P}}{\sup }{\mathbb{E}}_P[E_{\tau }]\le 1,E_{\tau }\ge 0.$$

E-processes are a strict superset of supermartingales. For example, the universal inference e-process is not a supermartingale. However, e-processes can be equivalently defined in terms of test martingales: $(E_t)$ is an e-process if there exists a family of test martingale $(M_t^P)$, $P\in \mathcal{P}$ such that

$$E_t\le M_t^P,\text{for all }P\in \mathcal{P}\text{ and }t\ge 0.$$

This representation implies that Ville’s inequality applies to e-processes. (In fact, the generalization of Ville’s inequality to compose distributions requires e-processes, see Ruf et al. (2023)). It also implies that e-processes always have the form

$$E_t=\underset{P}{\inf }{M}_t^P.$$

The game-theoretic interpretation (game-theoretic statistics) is to partition to null, play a game on each, and take your overall wealth to be the minimum in each game.

Like e-values, e-processes are measures of evidence against the null, but in sequential settings.