Abstract

While empirical Bayes methods thrive in the presence of the thousands of simultaneous hypothesis tests in genomics and other large-scale applications, significance tests and confidence intervals are considered more appropriate for small numbers of tested hypotheses. Indeed, for fewer hypotheses, there is more uncertainty in empirical Bayes estimates of the prior distribution. Confidence intervals have been used to propagate the uncertainty in the prior to empirical Bayes inference about a parameter, but only by combining a Bayesian posterior distribution with a confidence distribution. Combining distributions of both types has also been used to combine empirical Bayes methods and confidence intervals for estimating a parameter of interest.

To clarify the foundational status of such combinations, the concept of an evidential model is proposed. In the framework of evidential models, both Bayesian posterior distributions and confidence distributions are special cases of evidential support distributions. Evidential support distributions, by quantifying the sufficiency of the data as evidence, leverage the strengths of Bayesian posterior distributions and confidence distributions for cases in which each type performs well and for cases benefiting from the combination of both. Evidential support distributions also address problems of bioequivalence, bounded parameters, and the lack of a unique confidence distribution.

摘要

虽然经验贝叶斯方法在基因组学和其他大规模应用中存在数千个同时假设检验的情况下蓬勃发展，但显着性检验和置信区间被认为更适合少量检验假设。实际上，对于较少的假设，先验分布的经验贝叶斯估计存在更多的不确定性。置信区间已用于传播关于参数的经验贝叶斯推断之前的不确定性，但只能通过将贝叶斯后验分布与置信分布相结合。结合这两种类型的分布也已用于结合经验贝叶斯方法和置信区间来估计感兴趣的参数。

为了阐明这种组合的基础地位，提出了证据模型的概念。在证据模型的框架中，贝叶斯后验分布和置信度分布都是证据支持分布的特例。证据支持分布通过将数据的充分性量化为证据，利用贝叶斯后验分布和置信分布的优势来处理每种类型表现良好的情况以及受益于两者结合的情况。证据支持分布还解决了生物等效性、有界参数和缺乏唯一置信分布的问题。