Estimating the effective sample size (ESS) of a prior distribution is an age-old yet pivotal challenge, with great implications for clinical trials and various biomedical applications. Although numerous endeavors have been dedicated to this pursuit, most of them neglect the likelihood context in which the prior is embedded, thereby considering all priors as "beneficial". In the limited studies of addressing harmful priors, specifying a baseline prior remains an indispensable step. In this paper, by means of the elegant bridge between the p-value and the posterior probability of the null hypothesis, we propose a new ESS estimation method based on p-value in the framework of hypothesis testing, expanding the scope of existing ESS estimation methods in three key aspects: (i) We address the specific likelihood context of the prior, enabling the possibility of negative ESS values in case of prior-likelihood disconcordance; (ii) By leveraging the well-established bridge between the frequentist and Bayesian configurations under noninformative priors, there is no need to specify a baseline prior which incurs another criticism of subjectivity; (iii) By incorporating ESS into the hypothesis testing framework, our $p$-value ESS estimation method transcends the conventional one-ESS-one-prior paradigm and accommodates one-ESS-multiple-priors paradigm, where the sole ESS may reflect the collaborative impact of multiple priors in diverse contexts. Through comprehensive simulation analyses, we demonstrate the superior performance of the p-value ESS estimation method in comparison with existing approaches. Furthermore, by applying this approach to an expression quantitative trait loci (eQTL) data analysis, we show the effectiveness of informative priors in uncovering gene eQTL loci.

翻译：有效样本量（ESS）的先验分布估计是一个古老而关键的问题，对临床试验和各种生物医学应用具有重要意义。尽管已有大量研究致力于此，但大多数忽略了先验所处的似然背景，从而将所有先验视为“有益的”。在为数不多的处理有害先验的研究中，指定基线先验仍是不可或缺的步骤。本文通过p值与零假设后验概率之间的优雅桥梁，在假设检验框架下提出了一种基于p值的新ESS估计方法，从三个关键方面拓展了现有ESS估计方法的范畴：（i）我们考虑了先验的特定似然背景，使得在先验-似然不一致时可能出现负ESS值；（ii）通过利用无信息先验下频率学派与贝叶斯配置之间已确立的桥梁，无需指定基线先验，从而避免了主观性争议；（iii）通过将ESS纳入假设检验框架，我们的$p$值ESS估计方法超越了传统“一个ESS对应一个先验”的范式，兼容“一个ESS对应多个先验”的范式，其中单一ESS可反映多个先验在不同情境下的协同影响。通过全面的模拟分析，我们证明了p值ESS估计方法相较于现有方法的优越性能。此外，通过将该方法应用于表达数量性状基因座（eQTL）数据分析，我们展示了信息性先验在发现基因eQTL位点方面的有效性。