Interval estimation is one of the most frequently used methods in statistical science, employed to provide a range of credible values a parameter is located in after taking into account the uncertainty in the data. However, while this interpretation only holds for Bayesian interval estimates, these suffer from two problems. First, Bayesian interval estimates can include values which have not been corroborated by observing the data. Second, Bayesian interval estimates and hypothesis tests can yield contradictory conclusions. In this paper a new theory for Bayesian hypothesis testing and interval estimation is presented. A new interval estimate is proposed, the Bayesian evidence interval, which is inspired by the Pereira–Stern theory of the full Bayesian significance test (FBST). It is shown that the evidence interval is a generalization of existing Bayesian interval estimates, that it solves the problems of standard Bayesian interval estimates and that it unifies Bayesian hypothesis testing and parameter estimation. The Bayesian evidence value is introduced, which quantifies the evidence for the (interval) null and alternative hypothesis. Based on the evidence interval and the evidence value, the (full) Bayesian evidence test (FBET) is proposed as a new, model-independent Bayesian hypothesis test. Additionally, a decision rule for hypothesis testing is derived which shows the relationship to a widely used decision rule based on the region of practical equivalence and Bayesian highest posterior density intervals and to the e-value in the FBST. In summary, the proposed method is universally applicable, computationally efficient, and while the evidence interval can be seen as an extension of existing Bayesian interval estimates, the FBET is a generalization of the FBST and contains it as a special case. Together, the theory developed provides a unification of Bayesian hypothesis testing and interval estimation and is made available in the R package fbst.

中文翻译：

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证据区间和贝叶斯证据值：关于贝叶斯假设检验和区间估计的统一理论

区间估计是统计科学中最常用的方法之一，在考虑数据的不确定性后，用于提供参数所在的一系列可信值。然而，虽然这种解释仅适用于贝叶斯区间估计，但它们存在两个问题。首先，贝叶斯区间估计可以包括尚未通过观察数据证实的值。其次，贝叶斯区间估计和假设检验会得出相互矛盾的结论。本文提出了贝叶斯假设检验和区间估计的新理论。提出了一种新的区间估计，贝叶斯证据区间，其灵感来自完全贝叶斯显着性检验 (FBST) 的 Pereira-Stern 理论。结果表明，证据区间是现有贝叶斯区间估计的推广，解决了标准贝叶斯区间估计的问题，统一了贝叶斯假设检验和参数估计。引入了贝叶斯证据值，它量化了（区间）零假设和备择假设的证据。基于证据区间和证据值，（全）贝叶斯证据检验（FBET）被提出作为一种新的、与模型无关的贝叶斯假设检验。此外，推导了假设检验的决策规则，该规则显示了与基于实际等价区域和贝叶斯最高后验密度区间的广泛使用的决策规则以及与 FBST 中的 e 值的关系。总之，所提出的方法是普遍适用的、计算效率高的，虽然证据区间可以看作是现有贝叶斯区间估计的扩展，但 FBET 是 FBST 的推广，并将其作为特例包含在内。总之，所开发的理论提供了贝叶斯假设检验和区间估计的统一，并在 R 包中提供 FBET 是 FBST 的推广，并将其作为特例包含在内。总之，所开发的理论提供了贝叶斯假设检验和区间估计的统一，并在 R 包中提供 FBET 是 FBST 的推广，并将其作为特例包含在内。总之，所开发的理论提供了贝叶斯假设检验和区间估计的统一，并在 R 包中提供fbst。