Extend e-processes from filtrations to arbitrary events beyond simple hypotheses

Determine whether e-processes can be generalized from being defined on filtrations (i.e., sequences of sigma-algebras associated with random variables) to being defined on arbitrary events of the sample space, in order to satisfy intra-experiment counterfactual invariance beyond the simple-versus-simple case where both the null hypothesis H0 and the alternative hypothesis H1 are singleton model classes.

Background

The paper distinguishes inter- and intra-experiment counterfactual desiderata for evidence measures. Likelihood ratios can be defined as functions on arbitrary events, which helps satisfy intra-experiment counterfactual invariance. In contrast, e-processes are typically defined on filtrations (sequences of sigma-algebras), not arbitrary events.

The authors note that while likelihood ratios induce e-processes in the simple-versus-simple setting, it is unresolved whether a comparable event-based formulation exists for e-processes in more general (e.g., composite) settings. Establishing such an extension would address a gap in meeting the intra-experiment counterfactual desideratum and strengthen connections to the likelihood principle.

References

On the other hand, one invariably defines e-processes on filtrations (essentially sequences of random variables), not on arbitrary events, and at the time of writing it is not clear whether they can be extended to events beyond the simplest case in which H0 and H1 are simple (this is also the reason we only marked the strict likelihood principle to be only approximately satisfied by e-processes — likelihood ratios are defined on arbitrary events, and e-processes on the more restricted notion of filtrations). Further work is required to answer this question.

— E-values as statistical evidence: A comparison to Bayes factors, likelihoods, and p-values  (2603.24421 - Chugg et al., 25 Mar 2026) in Intra-experiment counterfactuals, Dynamic Criteria, Section 5 (Comparing Evidence Measures)