In the hypothesis testing framework, $p$‑value is often computed to determine whether to reject the null hypothesis or not. On the other hand, Bayesian approaches typically compute the posterior probability of the null hypothesis to evaluate its plausibility. We revisit Lindley’s paradox and demystify the conflicting results between Bayesian and frequentist hypothesis testing procedures by casting a two-sided hypothesis as a combination of two one-sided hypotheses along the opposite directions. This formulation can naturally circumvent the ambiguities of assigning a point mass to the null and choices of using local or non-local prior distributions. As $p$‑value solely depends on the observed data without incorporating any prior information, we consider non-informative prior distributions for fair comparisons with $p$‑value. The equivalence of $p$‑value and the Bayesian posterior probability of the null hypothesis can be established to reconcile Lindley’s paradox. More complicated settings, such as multivariate cases, random effects models and non-normal data, are also explored for generalization of our results to various hypothesis tests.

中文翻译：

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通过连接 $p$-value 和后验概率揭开 Lindley 悖论的神秘面纱

在假设检验框架中，通常计算 $p$-value 以确定是否拒绝原假设。另一方面，贝叶斯方法通常计算原假设的后验概率以评估其合理性。我们重新审视林德利悖论，并通过将双边假设作为沿相反方向的两个单边假设的组合来揭开贝叶斯和频率论假设检验程序之间相互矛盾的结果的神秘面纱。这种公式可以自然地避免将点质量分配给零点以及选择使用局部或非局部先验分布的歧义。由于 $p$-value 仅取决于观察到的数据而不包含任何先验信息，我们考虑非信息性先验分布以与 $p$-value 进行公平比较。可以建立 $p$-value 的等价性和原假设的贝叶斯后验概率来调和 Lindley 悖论。还探索了更复杂的设置，例如多变量案例、随机效应模型和非正态数据，以将我们的结果推广到各种假设检验。