The practice of sequentially testing a null hypothesis as data are collected until the null hypothesis is rejected is known as optional stopping. It is well known that optional stopping is problematic in the context of p value-based null hypothesis significance testing: The false-positive rates quickly overcome the single test’s significance level. However, the state of affairs under null hypothesis Bayesian testing, where p values are replaced by Bayes factors, has perhaps surprisingly been much less consensual. Rouder (2014) used simulations to defend the use of optional stopping under null hypothesis Bayesian testing. The idea behind these simulations is closely related to the idea of sampling from prior predictive distributions. Deng et al. (2016) and Hendriksen et al. (2020) have provided mathematical evidence to the effect that optional stopping under null hypothesis Bayesian testing does hold under some conditions. These papers are, however, exceedingly technical for most researchers in the applied social sciences. In this paper, we provide some mathematical derivations concerning Rouder’s approximate simulation results for the two Bayesian hypothesis tests that he considered. The key idea is to consider the probability distribution of the Bayes factor, which is regarded as being a random variable across repeated sampling. This paper therefore offers an intuitive perspective to the literature and we believe it is a valid contribution towards understanding the practice of optional stopping in the context of Bayesian hypothesis testing.

在收集数据时顺序测试零假设直到拒绝零假设的做法称为可选停止。众所周知，在基于p值的零假设显着性检验的情况下，可选停止是有问题的：假阳性率很快超过了单一检验的显着性水平。然而，零假设贝叶斯检验下的事态，其中p值被贝叶斯因子取代，也许令人惊讶的是，共识要少得多。Rouder (2014) 使用模拟来捍卫在零假设贝叶斯测试下使用可选停止。这些模拟背后的想法与从先前预测分布中抽样的想法密切相关。邓等人。（2016）和亨德里克森等人。(2020) 提供了数学证据，表明在某些条件下，零假设贝叶斯检验下的可选停止确实成立。然而，这些论文对于应用社会科学研究的大多数研究人员来说是非常技术性的。在本文中，我们提供了关于 Rouder 对他考虑的两个贝叶斯假设检验的近似模拟结果的一些数学推导。关键思想是考虑贝叶斯因子的概率分布，它被认为是重复抽样中的随机变量。因此，本文为文献提供了一个直观的视角，我们相信它对理解贝叶斯假设检验背景下的可选停止实践是一个有效的贡献。