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

TL;DR

This paper introduces a Bayesian method with ignorance priors for detecting Lorentzian profiles in power spectra, providing more reliable parameter estimates and uncertainty quantification than traditional maximum likelihood approaches.

Contribution

It develops a conservative Bayesian framework using MCMC to assess the presence of Lorentzian profiles in power spectra with minimal assumptions, improving mode detection accuracy.

Findings

The Bayesian approach yields more accurate probability distributions for parameters.

It avoids assumptions inherent in maximum likelihood estimation.

Applied to CoRoT data, it demonstrated improved detection of stellar pulsation modes.

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 observations of the solar-type pulsator HD 49933.