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

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 solartype pulsation in stars other than the Sun where the observations are of lower quality and can be over-interpreted. As an example, we

[1]  Steven Dewitte,et al.  In-Flight Performance of the Virgo Solar Irradiance Instruments on Soho , 1997 .

[2]  W. Chaplin,et al.  The observation and simulation of stochastically excited solar p modess , 1997 .

[3]  Alyson G. Wilson From solar min to max: half a solar cycle with SOHO , 2002 .

[4]  P. Gregory Bayesian Logical Data Analysis for the Physical Sciences: The how-to of Bayesian inference , 2005 .

[5]  P. C. Gregory A Bayesian Kepler periodogram detects a second planet in HD 208487 , 2006 .

[6]  Brendon J. Brewer,et al.  BAYESIAN INFERENCE FROM OBSERVATIONS OF SOLAR-LIKE OSCILLATIONS , 2007 .

[7]  B. Croll Markov Chain Monte Carlo Methods Applied to Photometric Spot Modeling , 2006 .

[8]  P. Gregory Bayesian Logical Data Analysis for the Physical Sciences: A Comparative Approach with Mathematica® Support , 2005 .

[9]  H. Kjeldsen,et al.  CoRoT sounds the stars: p-mode parameters of Sun-like oscillations on HD 49933 , 2008 .

[10]  L. Gizon,et al.  The art of fitting p-mode spectra - I. Maximum likelihood estimation , 1997 .

[11]  S. Ravi Bayesian Logical Data Analysis for the Physical Sciences: a Comparative Approach with Mathematica® Support , 2007 .

[12]  W. K. Hastings,et al.  Monte Carlo Sampling Methods Using Markov Chains and Their Applications , 1970 .

[13]  Philip C. Gregory,et al.  Bayesian Logical Data Analysis for the Physical Sciences: Acknowledgements , 2005 .

[14]  T.Appourchaux Bayesian approach for g-mode detection, or how to restrict our imagination , 2007, 0711.0435.

[15]  Laurent Gizon,et al.  Determining the Inclination of the Rotation Axis of a Sun-like Star , 2003 .

[16]  M. Auvergne,et al.  Intrinsic photometric characterisation of stellar oscillations and granulation Solar reference values and CoRoT response functions , 2008, 0809.1078.

[17]  A. Gelman,et al.  Weak convergence and optimal scaling of random walk Metropolis algorithms , 1997 .

[18]  L. Gizon,et al.  The art of fitting p-mode spectra , 1998 .

[19]  Cambridge,et al.  Characterising stellar micro-variability for planetary transit searches , 2003, astro-ph/0310381.