Bayesian Inversionof 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)

Outline

• Introduction

• Bayesian Inversion

• Markov Chain Monte Carlo

• Applications

• Conclusions

Introduction

ObservationsModel

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?)

Introduction

If we are living in a perfect, ideal, Teletubbie world with

Nonoise

Noambiguities

Nodegeneracies

There will be ONE model that better explains the observations and onecan safely say that this is THE correct model

Introduction

However, we are fortunately living in an imperfect, non ideal world with

Noise Ambiguities Degeneracies

There will be more than ONE model that better explains the observations and one cannot say that only one model is THE correct model

Introduction

As a consequence, any inversion procedure carried out in our noisy andambiguous 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 aprobabilistic problem that has to be tackled using a statistical approach

We give the probability that any given model explains the observables

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

Bayesian Inference

The inductive inference problem is to update from our a-priori knowledge ofthe parameters to a a-posteriori knowledge after taking into account the

information encoded in the observed dataset

Bayes theorem

Posterior probability

Prior probabilityLikelihood

Priors

Any Bayesian reasoning scheme introduces the prior probability (a-prior information)

Typical priorsTop-hat function(flat prior)

imaxmin

Gaussian prior(we know a highprobable value)

i

Assuming statistical independence for all parameters the total prior can be calculated as

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

Bayesian Inference

In order to completely solve the inversion procedure, we NEED to knowthe 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

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 MN evaluations of the forward problemto obtain the full posterior distribution

Example: if N=10 (typical for ME models) and M=10 (relatively coarse grid), we end up with

1010 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!!!

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 withthe number of parameters)

MCMC. Technical details

Propose an initialset of parameters 0

Calculate the posteriorp(0|D)

Obtain new set of parameters

sampling from q(i|i-1)

Calculate the posteriorp(i|D) and the ratio

Accept set of parameters i with probability

Bayesian Inference. Simple example

Andrieu et al. 2003

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 distributionbut 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

MCMC. Possible post-facto analysis

If chains start far from the region of large posterior probability, it takes someiterations to locate the region the first Nburn-in iterations are thrown away

BURN-IN

Starting point

Burn-in

Reduces the size of the chain hopefully maintaining their propertiesLess used now due to the increase of the computer capabilities

THINNING

Academic example =10-5 Ic

Original valuesB=100 GB=45º

Academic example =10-5 Ic

Markov chains without burn-in and thinning

Marginalization (multi-dimensionalintegration) is obtained by “making

histograms”

Academic example =10-3 Ic

Original valuesB=100 GB=45º

B cos B=cte

Academic example =10-3 Ic

Academic example =10-3 Ic

B cos B=100 cos 45º = 70.7 G

Realistic example =10-4 IcStokes profiles with a low flux (10 Mx/cm2)

B=1000 Gf = 1%

Fields between 500 and 1800 G are compatible withthe observed Stokes profiles at 1 confidence level

Some parameters are not constrained by the observables

Low flux region

The “thermodynamical” parameters of thenon-magnetic component can be nicely

constrained by the data

Low flux region

High flux region =10-4 IcStokes profiles with a high flux (200 Mx/cm2)

B=1000 Gf = 20%

Magnetic field strength and other parameters of the magnetic

component are constrained by the data

High flux region =10-4 Ic

Broader confidence levels are seenin the non-magnetic component due to

its reduced filling factor

Lack of information

Marginalized magnetic field strength posterior distribution

Note the similarity with the prior distribution

Observed sunspot

Umbral profile observed with THÉMIS

6302 Å

Observed sunspot

Umbral profile observed with THÉMIS

6301 Å

Combination of information

Inclusion of new information is trivial underthe Bayesian approach by directly

multiplying their posteriors

The 6302 Å line constrains better themagnetic field vector than the 6301 Å

line in this case

6301 Å

6302 Å

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