Title: E is for Evidence

Abstract: 

How much evidence do the data give us about one hypothesis versus  another? The standard way to measure evidence is still the p-value,   despite a myriad of problems surrounding it.  We present the e-value, a  recently popularized notion of  evidence which overcomes some of these  issues. While e-values were only given a name as recently as 2019, interest in them  has since exploded with papers in the Annals, JRSS B, Biometrika and the like – June 2022 saw the first international workshop on e-values, a second one is planned.
In  simple cases, e-values coincide with Bayes factors. But if the null is  composite or nonparametric, or an alternative cannot be explicitly  formulated, e-values and Bayes factors become distinct and e-processes  can be seen as a generalization of nonnegative supermartingales. Thus, unlike  the Bayes factor, e-values always allow for tests with strict frequentist  Type-I error control under optional continuation of data collection and combination of data  from different sources. E-values are also the basic building blocks of anytime-valid confidence intervals  that remain valid under continuous monitoring and optional stopping. In parametric settings they tend to be strictly wider than, hence consistent with Bayesian credible intervals. This led to the development of the e-posterior, an analogue to the Bayesian posterior that gets wider rather than wrong if the prior is chosen badly.
This work is based on:
P. Grunwald,. R. de Heide, W. Koolen (2023). Safe Testing. To appear in J. Roy. Stat. Soc., Series B
P. Grunwald (2023) . The E-Posterior. Proc. Phil. Trans. Soc. London Series A, 2023.