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