1. 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

2. Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009 Contents Bayesian over frequentists How does a Bayesian approach work? Results for asteroseismology Conclusion

3. Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009 Frequentist (MLE) Parameters to be evaluated (frequency,…) Data (power spectra,…) Information (a priori) Likelihood

4. Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009 Frequentist (MLE)

5. 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

6. 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)

7. 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)

8. 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

9. 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

10. 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

11. Symposium CoRoT 2009 – Cité Universitaire – 2 February 2009 MLE: –Garcia et alPoster P-II-016 –Deheuvels et alPoster 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

12. 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