On posterior probability and significance level: application to the power spectrum of HD49933 observed by CoRoT

We emphasize that the mention of the significance level when rejecting the null hypothesis (H0) which assumes that what is observed is pure noise) can mislead one to think that the H0 hypothesis is unlikely to occur with that significance level. We show that the significance level has nothing to do with the posterior probability of H0 given the observed data set, and that this posterior probability is very much higher than what the significance level naively provides. We use Bayes theorem for deriving the posterior probability of H0 being true assuming an alternative hypothesis H1 that assumes that a mode is present, taking some prior for the mode height, for the mode amplitude and linewidth.We report the posterior probability of H0 for the p modes detected on HD49933 by CoRoT. We conclude that the posterior probability of H0 provide a much more conservative quantification of the mode detection than the significance level. This framework can be applied to any stellar power spectra similar to those obtained for asteroseismology.

💡 Research Summary

The paper addresses a common misconception in asteroseismic data analysis: the belief that a low significance level (p‑value) directly reflects a low probability that the null hypothesis (H₀, pure noise) is true. The authors clarify that a p‑value is the probability of obtaining data at least as extreme as the observed, assuming H₀ is correct; it does not quantify the posterior probability of H₀ given the data. To obtain a genuine measure of confidence in a detection, they apply Bayes’ theorem, explicitly modeling both H₀ and an alternative hypothesis (H₁) that a stellar oscillation mode is present.

The Bayesian framework requires priors for the physical parameters describing a mode: height (or power), amplitude, and linewidth. The authors adopt physically motivated prior distributions—exponential for mode height, log‑normal for amplitude, and uniform for linewidth—reflecting typical expectations for solar‑like oscillators. The likelihood for each frequency bin of the power spectrum follows a χ² distribution with two degrees of freedom, appropriate for a Fourier power spectrum of Gaussian noise. Under H₀ the likelihood is simply an exponential decay with the background noise level σ; under H₁ the likelihood incorporates a Lorentzian profile characterized by the mode parameters, integrated over their priors.

Bayes’ theorem then yields the posterior probability of the null hypothesis: \