The look-elsewhere effect from a unified Bayesian and frequentist perspective

TL;DR

This paper introduces a unified Bayesian and frequentist approach to accurately quantify the look-elsewhere effect, enabling fast and reliable significance assessment of anomalies across large parameter spaces without costly simulations.

Contribution

It develops a continuous correction method using Laplace approximation, relating the trials factor to prior-posterior volume ratios, and generalizes Wilks' theorem for model complexities.

Findings

Works well across full p-value range, including non-asymptotic regimes.

Provides a fast, simulation-free way to account for the look-elsewhere effect.

Naturally incorporates additional model degrees of freedom.

Abstract

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.