  Multi-parameter Bayesian inferences based on Markov chain Monte Carlo (MCMC)
samples have been widely used to estimate red noise in solar period-periodic
signals. To MCMC, proper priors and sufficient iterations are prerequisites
ensuring the accuracy of red noise estimation. We used MCMC-based Bayesian
inferences to estimate 100 groups of red noise synthesized randomly for
evaluating its accuracy. At the same time, the Brooks-Gelman algorithm was
employed to precisely diagnose the convergence of the Markov chains generated
by MCMC. The root-mean-square error of parameter inferences to the synthetic
data is only 1.14. Furthermore, we applied the algorithm to analyze the
oscillation modes in a sunspot and a flare. A 70 s period is detected in the
sunspot umbra in addition to 3- and 5-minute periods, and a 40 s period is
detected in the flare. The results prove that estimating red noise with
MCMC-based Bayesian has more high accuracy in the case of proper priors and
convergence. We also find that the number of iterations increases dramatically
to achieve convergence as the number of parameters grows. Therefore, we
strongly recommend that when estimating red noise with MCMC-based Bayesian,
different initial values must be selected to ensure that the entire posterior
distribution is covered.
