When searching over a large parameter space for anomalies such as events, peaks, objects, or particles, there is a large probability that spurious signals with seemingly high significance will be found. This is known as the look-elsewhere effect and is prevalent throughout cosmology, (astro)particle physics, and beyond. To avoid making false claims of detection, one must account for this effect when assigning the statistical significance of an anomaly. This is typically accomplished by considering the trials factor, which is generally computed numerically via potentially expensive simulations. In this paper we develop a continuous generalization of the Bonferroni and Sidak corrections by applying the Laplace approximation to evaluate the Bayes factor, and in turn relating the trials factor to the prior-to-posterior volume ratio. We use this to define a test statistic whose frequentist properties have a simple interpretation in terms of the global $p$-value, or statistical significance. We apply this method to various physics-based examples and show it to work well for the full range of $p$-values, i.e. in both the asymptotic and non-asymptotic regimes. We also show that this method naturally accounts for other model complexities such as additional degrees of freedom, generalizing Wilks' theorem. This provides a fast way to quantify statistical significance in light of the look-elsewhere effect, without resorting to expensive simulations.

中文翻译：

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从统一的贝叶斯和频率论角度看别处效应

在大型参数空间中搜索事件、峰值、对象或粒子等异常时，很有可能会发现看似重要的虚假信号。这被称为“别处看”效应，在整个宇宙学、（天体）粒子物理学等领域都很普遍。为了避免对检测做出虚假声明，在分配异常的统计显着性时必须考虑到这种影响。这通常是通过考虑试验因素来实现的，试验因素通常通过可能昂贵的模拟进行数值计算。在本文中，我们通过应用拉普拉斯近似来评估贝叶斯因子，进而将试验因子与前后体积比相关联，对 Bonferroni 和 Sidak 校正进行了连续推广。我们用它来定义一个测试统计量，它的常客属性在全局 $p$-value 或统计显着性方面有一个简单的解释。我们将此方法应用于各种基于物理学的示例，并表明它在所有 $p$ 值范围内都能很好地工作，即在渐近和非渐近状态下。我们还表明，这种方法自然地解释了其他模型的复杂性，例如额外的自由度，概括了威尔克斯定理。这提供了一种根据外观别处效应量化统计显着性的快速方法，而无需求助于昂贵的模拟。我们将此方法应用于各种基于物理学的示例，并表明它在所有 $p$ 值范围内都能很好地工作，即在渐近和非渐近状态下。我们还表明，这种方法自然地解释了其他模型的复杂性，例如额外的自由度，概括了威尔克斯定理。这提供了一种根据外观别处效应量化统计显着性的快速方法，而无需求助于昂贵的模拟。我们将此方法应用于各种基于物理学的示例，并表明它在所有 $p$ 值范围内都能很好地工作，即在渐近和非渐近状态下。我们还表明，这种方法自然地解释了其他模型的复杂性，例如额外的自由度，概括了威尔克斯定理。这提供了一种根据外观别处效应量化统计显着性的快速方法，而无需求助于昂贵的模拟。