The Jeffreys-Lindley paradox exposes a rift between Bayesian and frequentist hypothesis testing that strikes at the heart of statistical inference. Contrary to what most current literature suggests, the paradox was central to the Bayesian testing methodology developed by Sir Harold Jeffreys in the late 1930s. Jeffreys showed that the evidence against a point-null hypothesis $\mathcal{H}_0$ scales with $\sqrt{n}$ and repeatedly argued that it would therefore be mistaken to set a threshold for rejecting $\mathcal{H}_0$ at a constant multiple of the standard error. Here we summarize Jeffreys's early work on the paradox and clarify his reasons for including the $\sqrt{n}$ term. The prior distribution is seen to play a crucial role; by implicitly correcting for selection, small parameter values are identified as relatively surprising under $\mathcal{H}_1$. We highlight the general nature of the paradox by presenting both a fully frequentist and a fully Bayesian version. We also demonstrate that the paradox does not depend on assigning prior mass to a point hypothesis, as is commonly believed.
翻译:Jeffreys-Lindley悖论揭示了巴伊西亚人和经常假设检验之间的裂痕,这种误差击中了统计推论的核心。与大多数文献所显示的相反,这种矛盾是哈罗德·杰弗里爵士在1930年代后期制定的巴伊西亚检验方法的核心。杰弗里斯爵士表明,针对一个点核假设的证据是美元=mathcal{H ⁇ 0美元,使用美元=sqrt{H ⁇ 0美元。我们反复争辩说,因此,在标准错误的多个常数中设定一个拒绝$\mathcal{H ⁇ 0美元的门槛是错误的。我们在这里总结了杰弗里斯关于悖论的早期工作,并澄清了他将$=sqrt{n}任期包括在内的理由。先前的分布被认为发挥着关键作用;通过暗含对选择的纠正,小参数值在$\mathcal{H ⁇ 1美元下被确定为相对令人惊讶的。我们强调这一悖论的一般性质,即提出一个完全频繁的版本和完全的巴伊西亚人的版本。我们还表明,这一悖论并不取决于将前半点假设划为通常认为的假设。