Luigi Pace and Alessandra Salvan’s contribution to the Discussion of ‘Safe testing’ by Grünwald, de Heide, and Koolen

原文摘要

We wish to congratulate the authors on their important paper. In this contribution, we will focus on parametric models, where e-values provide a welcome new perspective on likelihood ratios and Bayes factors, as well as a useful addition to the discussion about the foundational concept of statistical evidence. The guarantees sought for in safe testing are frequentist, but prior information may be injected in Bayesian terms, establishing an alliance between traditionally antagonistic inferential paradigms. The core ideas of safe testing are that error probabilities need to be constrained only under H0⁠, and that their evaluation is conservative and nonasymptotic, relying on Ville’s inequality, which is the basis for anytime-valid inference (Ramdas et al., 2023) in the form of tests or confidence sequences. Incidentally, the associated safe bound on the error probabilities may deserve a new name, different from the traditional significance/confidence level. Perhaps, strong significance/confidence level, or persistence level, as suggested in Pace and Salvan (2020) for confidence sequences. Using a likelihood ratio threshold as a universal yardstick, likelihood-based e-values obey the strong likelihood principle, in spite of the opinion that any frequentist desideratum is irreconcilable with such principle. A consequence of obeying the strong likelihood principle is the validity of e-values under conditioning on ancillary statistics, because likelihood factors not depending on the parameter cancel. Ancillary conditioning is a rather Fisherian concern, but a resemblance with e-values may be highlighted for simple H0 and H1⁠. Indeed, in a fixed n setting, when the likelihood ratio r has continuous distributions, Pace and Salvan (2014) show that there exists an ancillary statistic a=a(r)⁠, i.e. a statistic having the same distribution under H0 and under H1 and which, joint with the maximum likelihood estimate, is minimal sufficient. As in safe testing, the conditional Type I and Type II error probabilities, α(a) and β(a)⁠, satisfy the universal inequalities α(a)≤1/a=1/r⁠, when r>1⁠, and β(a)≤s(a)=r⁠, when r<1⁠. Under likelihood ratio symmetry (Berger et al., 1994), i.e. when 1/r is distributed under H0 as r under H1⁠, it is seen that α(a)=1/(a+1)=1/(r+1)⁠, when r>1⁠, so that α(a)≤α entails r≥(1−α)/α⁠. We wonder whether substitution of the threshold 1/α with (1−α)/α might be meaningful also for tests based on e-values.

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