  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.
