  Many astrophysical sources, especially compact accreting sources, show
strong, random brightness fluctuations with broad power spectra in addition to
periodic or quasi-periodic oscillations (QPOs) that have narrower spectra. The
random nature of the dominant source of variance greatly complicates the
process of searching for possible weak periodic signals. We have addressed this
problem using the tools of Bayesian statistics; in particular using Markov
chain Monte Carlo techniques to approximate the posterior distribution of model
parameters, and posterior predictive model checking to assess model fits and
search for periodogram outliers that may represent periodic signals. The
methods developed are applied to two example datasets, both long XMM-Newton
observations of highly variable Seyfert 1 galaxies: RE J1034+396 and Mrk 766.
In both cases a bend (or break) in the power spectrum is evident. In the case
of RE J1034+396 the previously reported QPO is found but with somewhat weaker
statistical significance than reported in previous analyses. The difference is
due partly to the improved continuum modelling, better treatment of nuisance
parameters, and partly to different data selection methods.
