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Tutorial on Bayesian Techniques for Inference. Asensio Ramos Instituto de Astrofísica de Canarias. Outline. General introduction The Bayesian approach to inference Examples Conclusions. The Big Picture. Deductive Inference. Statistical Inference. The Big Picture. - PowerPoint PPT Presentation
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Tutorial onBayesian Techniques
for Inference
A. Asensio RamosInstituto de Astrofísica de Canarias
Outline
• General introduction
• The Bayesian approach to inference
• Examples
• Conclusions
The Big Picture
Predictions
ObservationData
Hypothesis testing
Parameter estimation
TestableHypothesis
(theory)
Statistical Inference
Deductive Inference
The Big Picture
Available information is always incomplete
Our knowledge of nature is necessarily probabilistic
Cox & Jaynes demonstrated that probability calculusfulfilling the rules
can be used to do statistical inference
Probabilistic inference
H1, H2, H3, …., Hn are hypothesis that we want to test
The Bayesian way is to estimate p(Hi|…) and selectdepending on the comparison of their probabilities
What are the p(Hi|…)???
But…
What is probability? (Frequentist)
In frequentist approach, probability describes “randomness”
If we carry out the experiment many times, which is the distribution ofevents (frequentist) p(x) is the histogram of random variable x
What is probability? (Bayesian)
In Bayesian approach, probability describes “uncertainty”
p(x) gives how probability is distributed among the possiblechoice of x
We observethis value
Everything can be a random variable as we will see later
Bayes theorem
It is trivially derived from the product rule
• Hi proposition asserting the truth of a hypothesis• I proposition representing prior information• D proposition representing data
Bayes theorem - Example
• Model M1 predicts a star at d=100 ly• Model M2 predicts a star at d=200 ly• Uncertainty in measurement is Gaussian with s=40 ly• Measured distance is d=120 ly
Likelihood
Posteriors
Bayes theorem – Another example
1.4% false negative(98.6% reliability)
2.3% false positive
Bayes theorem – Another example
H you have the diseaseH you don’t have the diseaseD1 your test is positive
You take the test and you get it positive. What is the probability that you have the disease if the incidence is 1:10000?
Bayes theorem – Another example
10-4 0.986
0.986 0.02310-4 0.9999
What is usually known as inversion
All inversion methods work by adjusting the parameters ofthe model with the aim of minimizing a merit function thatcompares observations with the synthesis from the model
One proposes a model to explain observations
Least-squares solution (maximum-likelihood) is the solution to theinversion problem
Defects of standard inversion codes
• Solution is given as a set of model parameters (max. likelihood)
• Not necessary the optimal solution• Sensitive to noise
• Error bars or confidence regions are scarce
• Gaussian errors• Not easy to propagate errors• Ambiguities, degeneracies, correlations are not detected
• Assumptions are not explicit
• Cannot compare models
Inversion as a probabilistic inference problem
Likelihood Prior
EvidencePosterior
Use Bayes theorem to propagate informationfrom data to our final state of knowledge
Priors
Typical priorsTop-hat function(flat prior)
qiqmaxqmin
Gaussian prior(we know some values
are more probablethan others)
qi
Assuming statistical independence for all parameters the total prior can be calculated as
Contain information about model parameters that we know before presenting the data
Likelihood
Assuming normal (gaussian) noise, the likelihood can be calculated as
where the c2 function is defined as usual
In this case, the c2 function is specific for the the case of Stokes profiles
Visual example of Bayesian inference
Advantages of Bayesian approach
• “Best fit” values of parameters are e.g., mode/median of the posterior
• Uncertainties are credible regions of the posterior
• Correlation between variables of the model are captured
• Generalized error propagation (not only Gaussian and including correl.)
Integration over nuissance parameters (marginalization)
Bayesian inference – an example
Hinode
Beautiful posterior distributions
Field strength Field inclination
Field azimuth Filling factor
Not so beautiful posterior distributions - degeneraciesField inclination
Inversion with local stray-light – be careful
si is the variance of the numerator
But… what happens if we propose a model like Orozco Suárez et al. (2007)with a stray-light contamination obtained from a local
average on the surrounding pixels
From observations
Variance becomes dependent on stray-light contamination
It is usual to carry out inversions with a stray-light contaminationobtained from a local average on the surrounding pixels
Spatial correlations: use global stray-light
It is usual to carry out inversions with a stray-light contaminationobtained from a local average on the surrounding pixels
If M correlations tend to zero
Spatial correlations
Lesson: use global stray-light contamination
Recommendation
Use global stray-light contaminationto avoid problems
But… the most general inversion method is…
Model comparison
Choose among the selected models the one that is preferred by the data
Posterior for model Mi
Model likelihood is just the evidence
Model comparison (compare evidences)
Model comparison – a worked example
H0 : simple GaussianH1 : two Gaussians of equal width but unknown amplitude ratio
H0 : simple Gaussian
H1 : two Gaussians of equal width but unknown amplitude ratio
Model comparison – a worked example
Model comparison – a worked example
Model comparison – a worked example
Model H1 is 9.2 times more probable
Model comparison – an example
Model 11 magnetic component
Model 21 magnetic+1 non-magnetic
component
Model 32 magnetic components
Model 42 magnetic components
with (v2=0, a2=0)
Model comparison – an example
Model 11 magnetic component
9 free parameters
Model 21 magnetic+1 non-magnetic
component17 free parameters
Model 32 magnetic components
20 free parameters
Model 42 magnetic components
with (v2=0, a2=0)18 free parameters
Model 2 is preferred by the data“Best fit with the smallest number of parameters”
Model averaging. One step further
Models {Mi, i=1..N} have a common subset of parameters y of interestbut each model depends on a different set of parameters q
or have different priors over these parameters
What all models have to say about parameters yAll of them give a “weighted vote”
Posterior for y including all models
Model averaging – an example
Hierarchical models
In the Bayesian approach, everything can be considered a random variable
DATAMODEL LIKELIHOOD
MARGINALIZATIONNUISANCE PAR.
PRIOR
INFERENCE
PRIORPAR.
Hierarchical models
In the Bayesian approach, everything can be considered a random variable
DATAMODEL LIKELIHOOD
MARGINALIZATIONNUISANCE PAR.
PRIOR
INFERENCE
PRIORPAR.
PRIOR PRIOR
Bayesian Weak-field
Bayes theorem
Advantage: everything is close to analytic
Bayesian Weak-field – Hierarchical priors
Priors depend on some hyperparameters over whichwe can again set priors and marginalize them
Bayesian Weak-field - Data
IMaX data
Bayesian Weak-field - Posteriors
Joint posteriors
Bayesian Weak-field - Posteriors
Marginal posteriors
Hierarchical priors - Distribution of longitudinal B
Hierarchical priors – Distribution of longitudinal B
We want to infer the distribution of longitudinal B frommany observed pixels taking into account uncertainties
Parameterize the distribution in terms of a vector a
Mean+varianceif Gaussian
Height of binsif general
Hierarchical priors – Distribution of longitudinal B
Hierarchical priors – Distribution of longitudinal B
We generate N synthetic profiles with
noise withlongitudinal field sampled from a
Gaussian distributionwith standard deviation
25 Mx cm-2
Hierarchical priors – Distribution of any quantity
Bayesian image deconvolution
Bayesian image deconvolution
PSF blurringusing linear expansion
Image is sparsein any basis
Maximum-likelihood solution (phase-diversity, MOMFBD,…)
Inference in a Bayesian framework
• Solution is given as a probability over model parameters
• Error bars or confidence regions can be easily obtained, including correlations, degeneracies, etc.
• Assumptions are explicit on prior distributions
• Model comparison and model averaging is easily accomplished
• Hierarchical model is powerful for extracting information from data
Hinode data
Continuum Total polarization
Asensio Ramos (2009)
Observations of Lites et al. (2008)
How much information? – Kullback-Leibler divergence
Field strength (37% larger than 1) Field inclination (34% larger than 1)
Measures “distance” between posterior and prior distributions
Posteriors
Field strength
Field azimuthField inclination
Stray-light
Field inclination – Obvious conclusion
Linear polarization is fundamental to obtain reliable inclinations
Field inclination – Quasi-isotropic
Isotropic field
Our prior
Field inclination – Quasi-isotropic
Representation
Marginal distribution for each parameter
Sample N values from the posterior and all values arecompatible with observations
Field strength – Representation
All maps compatible with observations!!!
Field inclination
All maps compatible with observations!!!
In a galaxy far far away… (the future)
RAWDATA
POSTERIOR+MARGINALIZATIONNON-IMPORTANT
PARAMETERS
INFERENCE
INSTRUMENTSWITH SYSTEMATICS
PRIORS
MODEL PRIORS
Conclusions
• Inversion is not an easy task and has to be considered as a probabilistic inference problem
• Bayesian theory gives us the tools for inference
• Expand our view of inversion as a model comparison/averaging problem (no model is the absolute truth!)
Thank youand be Bayesian, my friend!