Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009 Analysis of power spectra of Sun-like stars using a Bayesian Approach Thierry Appourchaux Symposium COROT 2009 – Cité Universitaire – 2 February 2009
Dec 18, 2015
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Analysis of power spectra of Sun-like stars using a Bayesian Approach
Thierry Appourchaux
Symposium COROT 2009 – Cité Universitaire – 2 February 2009
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Contents
• Bayesian over frequentists• How does a Bayesian approach work?• Results for asteroseismology• Conclusion
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Frequentist (MLE)
Parameters to be evaluated (frequency,…)
Data (power spectra,…)
Information (a priori)
Likelihood
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Frequentist (MLE)
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Bayes theorem
Posterior probability
Prior probability
Normalisation factor
Parameters to be evaluated (frequency,…)
Data (power spectra,…)
Information (a priori)
Likelihood
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Bayesian over frequentist
• Bayesians address the question everyone is interested in by using assumptions no-one believes. (i.e. the validity of the prior)
• Frequentists use impeccable logic to deal with an isssue of no interest to anyone. (i.e. the trueness of the likelihood)
Lyons (2007)
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Bayes theorem in practice
• A priori: any information known (statistics, parameters,…)• What ?
– the posterior probability of the parameters• How ?
– Set prior probability for the parameters knowing what you know (or believe to know…)
– Express likelihood of the data given the parameters– Recover Posterior probability by:
• In full using Markov Chains or• Maximizing the Posterior probability (MAP)
– Recover global likelihood of the model
Gregory (2005)
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Maximum A Posteriori (MAP)
Example with a Gaussian prior:
See Gaulme et al (MAP) Poster P-II-013
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Markov Chain Monte Carlo (MCMC)
• Primary objective: recover in full the posterior probability
• If analytical solution not feasible use numerical solution using MCMC based on the Metropolis-Hasting algorithm
• Explore space parameters with parallel tempering
• Provide either the full posterior probability or the median and the associated quartiles
• Recover global likelihood of the model
See Benomar et al (Markov Chains) Poster P-II-012
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Some results
Benomar et al (MCMC) Poster P-II-012
Gaulme et al (MAP) Poster P-II-013
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
• MLE:– Garcia et al Poster P-II-016– Deheuvels et al Poster P-II-018
• Bayes: – Reegen Poster P-I-005– Benomar et al (Markov Chains) Poster P-II-012 – Gaulme et al (MAP) Poster P-II-013– Campante et al (MAP) Poster P-II-015
Power spectrum fitting
Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009
Conclusion
• Bayesian approach is slowly starting• A lot work needed for exploring various alleys• Don’t forget: the Bayesian approach is more
conservative…• …and look at the posters