Stefano Rizzelli’s contribution to the Discussion of ‘Safe testing’ by Grünwald, de Heide, and Koolen
本文发展了基于e值的假设检验理论，e值能轻松合并多项研究结果，并在可选继续研究时保证第一类错误率，提出了增长最优性概念，并展示了如何构造GRO e变量。
We develop the theory of hypothesis testing based on the e-value, a notion of evidence that, unlike the p-value, allows for effortlessly combining results from several studies in the common scenario where the decision to perform a new study may depend on previous outcomes.Tests based on e-values are safe, i.e. they preserve type-I error guarantees, under such optional continuation.We define growth rate optimality (GRO) as an analogue of power in an optional continuation context, and we show how to construct GRO e-variables for general testing problems with composite null and alternative, emphasizing models with nuisance parameters.GRO e-values take the form of Bayes factors with special priors.We illustrate the theory using several classic examples including a 1-sample safe t-test and the 2 2 contingency table.Sharing Fisherian, Neymanian, and Jeffreys-Bayesian interpretations, e-values may provide a methodology acceptable to adherents of all three schools.