1. Bayesian Inversion of Stokes Profiles A.Asensio Ramos (IAC) M. J. Martínez González (LERMA) J. A. Rubiño Martín (IAC) Beaulieu Workshop ( Beaulieu sur Mer, 8-10 October 2007)

2. Outline Introduction Bayesian Inversion Markov Chain Monte Carlo Applications Conclusions

3. Introduction Observations Model What is an inversion process? We have a set of observations and we propose a physical model FORWARD PROBLEM (Typically univoque) INVERSION PROBLEM (is it univoque?)

4. Introduction If we are living in a perfect, ideal, Teletubbie world with No noise No ambiguities No degeneracies There will be ONE model that better explains the observations and one can safely say that this is THE correct model

5. Introduction However, we are fortunately living in an imperfect, non ideal world with NoiseAmbiguitiesDegeneracies There will be more than ONE model that better explains the observations and one cannot say that only one model is THE correct model

6. Introduction As a consequence, any inversion procedure carried out in our noisy and ambiguous world cannot give only one model as solution but has to give a set of models that are compatible with our observables Any inversion problem has to be understood as a probabilistic problem that has to be tackled using a statistical approach We give the probability that any given model explains the observables

7. Bayesian Inference D represents our observables (Stokes profiles) M represents our model (Milne-Eddington, LTE,...) and it is parameterized by a vector of parameters  We have some a-priori knowledge of their values

8. Bayesian Inference The inductive inference problem is to update from our a-priori knowledge of the parameters to a a-posteriori knowledge after taking into account the information encoded in the observed dataset Bayes theorem Posterior probability Prior probability Likelihood

9. Priors Any Bayesian reasoning scheme introduces the prior probability (a-prior information) Typical priors Top-hat function (flat prior) ii  max  min Gaussian prior (we know a high probable value) ii Assuming statistical independence for all parameters the total prior can be calculated as

10. Likelihood Assuming normal (gaussian) noise, the likelihood can be calculated as where the  2 function is defined as usual In this case, the  2 function is specific for the the case of Stokes profiles

11. Bayesian Inference In order to completely solve the inversion procedure, we NEED to know the complete p(  |D) posterior probability distribution Sometimes, we are interested only in a subset of parameters Marginalization In any case, we still need the complete posterior distribution

12. Bayesian Inference. The naïve approach Our model is parameterized with N parameters We use M values for each parameter (to have a good coverage) We end up with M N evaluations of the forward problem to obtain the full posterior distribution Example: if N=10 (typical for ME models) and M=10 (relatively coarse grid), we end up with 10 10 evaluations  ~31 years if each model is evaluated in 0.1 s Only one experiment is possible during a typical scientific life!!! You better choose the correct experiment!!!

13. Bayesian Inference. The practical approach Markov Chain Monte Carlo “HAPPY IDEA”!! Build a Markov Chain with an equilibrium probability distribution function equal to the posterior distribution GOOD NEWS!! The Markov Chain rapidly converges towards the desired distribution using a reduced amount of evaluations (typically increases linearly with the number of parameters)

14. MCMC. Technical details Propose an initial set of parameters  0 Calculate the posterior p(  0 |D) Obtain new set of parameters sampling from q(  i |  i-1 ) Calculate the posterior p(  i |D) and the ratio Accept set of parameters  i with probability 

15. Bayesian Inference. Simple example Andrieu et al. 2003

16. Bayesian Inference. Proposal density The proposal density is the key point in MCMC methods It should ideally be as similar as possible to the posterior distribution but easy to calculate Typical proposal densities Uniform distribution In the limit that the proposal is equal to the distribution you want to sample, all proposed models are accepted Multi-dimensional gaussian

17. MCMC. Possible post-facto analysis If chains start far from the region of large posterior probability, it takes some iterations to locate the region  the first N burn-in iterations are thrown away BURN-IN Starting point Burn-in Reduces the size of the chain hopefully maintaining their properties Less used now due to the increase of the computer capabilities THINNING

18. Academic example  =10 -5 I c Original values B=100 G  B =45º

19. Academic example  =10 -5 I c Markov chains without burn-in and thinning Marginalization (multi-dimensional integration) is obtained by “making histograms”

20. Academic example  =10 -3 I c Original values B=100 G  B =45º B cos  B =cte

21. Academic example  =10 -3 I c

22. B cos  B =100 cos 45º = 70.7 G

23. Realistic example  =10 -4 I c Stokes profiles with a low flux (10 Mx/cm 2 ) B=1000 G f = 1% Fields between 500 and 1800 G are compatible with the observed Stokes profiles at 1  confidence level Some parameters are not constrained by the observables

24. Low flux region The “thermodynamical” parameters of the non-magnetic component can be nicely constrained by the data

25. Low flux region

26. High flux region  =10 -4 I c Stokes profiles with a high flux (200 Mx/cm 2 ) B=1000 G f = 20% Magnetic field strength and other parameters of the magnetic component are constrained by the data

27. High flux region  =10 -4 I c Broader confidence levels are seen in the non-magnetic component due to its reduced filling factor

28. Inclined fields Only Stokes I and V have been considered until now What happens if linear polarization is also taken into account? Do we see an improvement? We consider three values of  B 20º  Linear polarization is very weak and it is below the noise level 70º  Circular polarization is very weak and it is below the noise level 45º  intermediate result

29. Inclined fields Posterior distributions are very similar to those of vertical fields Linear polarization appears but circular polarization amplitudes come closer to the noise level

30. Lack of information Marginalized magnetic field strength posterior distribution Note the similarity with the prior distribution

31. Observed sunspot Umbral profile observed with THÉMIS 6302 Å

32. Observed sunspot Umbral profile observed with THÉMIS 6301 Å

33. Combination of information Inclusion of new information is trivial under the Bayesian approach by directly multiplying their posteriors The 6302 Å line constrains better the magnetic field vector than the 6301 Å line in this case 6301 Å 6302 Å

34. Conclusions The inversion process is a statistical problem  give set of parameters of a model that are compatible with the observables for a given noise Bayesian methods allow us to move from the a-priori information to a-posteriori situation using the information encoded in the data The posterior and their marginalized distributions are easily obtained using a Markov Chain Monte Carlo method Applications to synthetic and real data show the potential of the technique and points out severe degeneration problems