Vladimir Vovk’s Contribution to The Discussion of ‘Testing by Betting: A Strategy for Statistical and Scientific Communication’ by Glenn Shafer
本文讨论了Shafer提出的下注分数（betting scores）与p值的关系，指出两者分别推广了Cournot原则，并引入e值概念；通过新例子说明下注分数能解决实际统计问题，且用e值替代p值可改善随机分割数据方法中的可重复性问题。
Glenn Shafer’s paper is a powerful appeal for a wider use of betting ideas and intuitions in statistics. He admits that p-values will never be completely replaced by betting scores, and I discuss it further in Vovk (2020a) (Appendix A) (one of the two online appendices that I have prepared to meet the word limit). Both p-values and betting scores generalise Cournot’s principle (Shafer, 2007), but they do it in their different ways, and both ways are interesting and valuable. The term ‘e-values’ emphasises the fundamental role of expectation in the definition of betting scores (somewhat similar to the role of probability in the definition of p-values). It appears that the natural habitat for ‘betting scores’ is game-theoretic while for ‘e-values’ it is measure-theoretic (Shafer, 2020); therefore, I will say ‘e-values’ in the online appendices (Vovk, 2020a,b), which are based on measure-theoretic probability. In the second online appendix (Vovk, 2020b), I give a new example showing that betting scores are not just about communication; they may allow us to solve real statistical and scientific problems (more examples are given in the comment by my co-author Ruodu Wang). David Cox (1975) discovered that splitting data at random not only allows flexible testing of statistical hypotheses but also achieves high efficiency. A serious objection to the method is that different people analysing the same data may get very different answers (thus violating ‘inferential reproducibility’, Goodman et al., 2016; Held & Schwab, 2020). Using e-values instead of p-values remedies the situation.