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In experiments where one searches a large parameter space for an anomaly, oneoften finds many spurious noise-induced peaks in the likelihood. This is knownas the look-elsewhere effect, and must be corrected for when performingstatistical analysis. This paper introduces a method to calibrate the falsealarm probability (FAP), or $p$-value, for a given dataset by considering theheights of the highest peaks in the likelihood. In the simplest form ofself-calibration, the look-elsewhere-corrected $\chi^2$ of a physical peak isapproximated by the $\chi^2$ of the peak minus the $\chi^2$ of the highestnoise-induced peak. Generalizing this concept to consider lower peaks providesa fast method to quantify the statistical significance with improved accuracy.In contrast to alternative methods, this approach has negligible computationalcost as peaks in the likelihood are a byproduct of every peak-search analysis.We apply to examples from astronomy, including planet detection, periodograms,and cosmology.

Self-Calibrating the Look-Elsewhere Effect: Fast Evaluation of the Statistical Significance Using Peak Heights