A Bayesian approach to power-spectrum significance estimation, with application to solar neutrino data

The usual procedure for estimating the significance of a peak in a power spectrum is to calculate the probability of obtaining that value or a larger value by chance, on the assumption that the time series contains only noise (e.g. that the measurements were derived from random samplings of a Gaussian distribution). However, it is known that one should regard this P-Value approach with caution. As an alternative, we here examine a Bayesian approach to estimating the significance of a peak in a power spectrum. This approach requires that we consider explicitly the hypothesis that the time series contains a periodic signal as well as noise. The challenge is to identify a probability distribution function for the power that is appropriate for this hypothesis. We propose what seem to be reasonable conditions to require of this function, and then propose a simple function that meets these requirements. We also propose a consistency condition, and check to see that our function satisfies this condition. We find that the Bayesian significance estimates are considerably more conservative than the conventional estimates. We apply this procedure to three recent analyses of solar neutrino data: (a) bimodality of GALLEX data; (b) power spectrum analysis of Super-Kamiokande data; and (c) the combined analysis of radiochemical neutrino data and irradiance data.