On the detection of Lorentzian profiles in a power spectrum: a Bayesian approach using ignorance priors

¹ Institute for Astronomy (IfA), University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, Austria e-mail: last name@astro.univie.ac.at
² Department of Astronomy and Physics, Saint Marys University, Halifax, NS B3H 3C3, Canada e-mail: guenther@ap.stmarys.ca

Received: 21 October 2008
Accepted: 11 August 2009

Abstract

Aims. Deriving accurate frequencies, amplitudes, and mode lifetimes from stochastically driven pulsation is challenging, more so, if one demands that realistic error estimates be given for all model fitting parameters. As has been shown by other authors, the traditional 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, show that a conservative Bayesian approach allows one to treat the detection of modes with minimal assumptions (i.e., about the existence and identity of the modes).

Methods. We derive a conservative Bayesian treatment for the probability of Lorentzian profiles being present in a power spectrum and describe an efficient implementation that evaluates the probability density distribution of parameters by using a Markov-chain Monte Carlo (MCMC) technique.

Results. Potentially superior to “best-fit” procedure like MLE, which only provides formal uncertainties, our method samples and approximates the actual probability distributions for all parameters involved. Moreover, it avoids shortcomings that make the MLE treatment susceptible to the built-in assumptions of a model that is fitted to the data. This is especially relevant when analyzing solar-type pulsation in stars other than the Sun where the observations are of lower quality and can be over-interpreted. As an example, we apply our technique to CoRoT (COnvection, ROtation and planetary Transits space mission, launched on 2006 December 27, was developed and is operated by the CNES, with participation of the Science Programs of ESA, ESAs RSSD, Austria, Belgium, Brazil, Germany and Spain.) observations of the solar-type pulsator HD 49933.