Bayesian Lorentzian profile fitting using Markov-Chain Monte Carlo: An observer’s approach

Aims. Investigating stochastically driven pulsation puts strong requirements on the quality of (observed) pulsation frequency spectra, such as the accuracy of frequencies, amplitudes, and mode life times and – important when fitting these parameters with models – a realistic error estimate which can be quite different to the formal error. As has been shown by other authors, the method of fitting Lorentzian profiles to the power spectrum of time-resolved photometric or spectroscopic data via the Maximum Likelihood Estimation (MLE) procedure delivers good approximations for these quantities. We, however, intend to demonstrate that a conservative Bayesian approach allows to treat this problem in a more consistent way. Methods. We derive a conservative Bayesian treatment for the probability of Lorentzian profiles being present in a power spectrum and describe its implementation via evaluating the probability density distribution of parameters by using the Markov-Chain Monte Carlo (MCMC) technique. In addition, we compare the results obtained by Appourchaux et al. (2008), who used the MLE technique, on the CoRoT data of HD 49933 to our Bayesian approach. Results. Rather than using a “best-fit” procedure like MLE, which can only deliver formal uncertainties, our procedure samples and approximates the actual probability distributions for all parameters involved. Moreover, it helps avoiding shortcomings that make the MLE treatment susceptible to the complexity of a model that is fitted to the data. This is especially relevant when analysing solar-type pulsation in other stars than the Sun where the observations usually have lower quality and we illustrate this claim in a reassessment of the pulsation of HD 49933.