  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 (known as the "p-value"), on the assumption that the time
series contains only noise - typically that the measurements are derived from
random samplings of a Gaussian distribution. We really need to know the
probability that the time series is - or is not - compatible with the null
hypothesis that the measurements are derived from noise. This probability can
be calculated by Bayesian analysis, but this requires one to specify and
evaluate a second hypothesis, that the time series does contain a contribution
other than noise. We approach the problem of identifying this function in two
ways. We first propose three simple conditions that it seems reasonable to
impose on this function, and show that these conditions may be satisfied by a
simple function with one free parameter. We then define two different ways of
combining information derived from two independent power estimates. We find
that this consistency condition may be satisfied, to good approximation, by a
special case of the previously proposed likelihood function. We find that the
resulting significance estimates are considerably more conservative than those
usually associated with the p-values. As two examples, we apply the new
procedure to two recent analyses of solar neutrino data: (a) power spectrum
analysis of Super-Kamiokande data, and (b) the combined analysis of
radiochemical neutrino data and irradiance data.
