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Applying Bayesian analysis to polarisation data

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

1 / 15


Outline

1 2 3 4

Polarisation simulations @ radio wavebands Bayesian analysis Analysis of polarisation data Conclusions

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

2 / 15


Radio sky simulations
Over view

Basic ingredients:
· Background/foreground polarised radio sources · Inter vening Faraday foreground screens

Adding complexity:
· Sources of polarised emission with intrinsic rotation

measures (radio halos and relics in LSS, galaxy disks and halos, AGN radio lobes etc.) Analysis challenge:
· Projection effects (tomography generally needs: prior

information; a large number of sources in the field) Large bandwidth, high sensitivity and high frequency and angular resolution are required
J. Geisbuesch (Cambridge) Bayesian analysis of polarisation data LOFAR MKSP Workshop 3 / 15


Radio sky simulations
Synthetic radio source sky

Recipe:
· Simulation of continuum

emission: S3 (Wilman'08) (extrapolated & evolved luminosity functions; source distribution) (populations: AGN types, normal & starburst galaxies)
· Intrinsic polarisation

modelling: Data: NVSS, ELIAS N1 etc. (empirically) Monte Carlo simulations of sources
J. Geisbuesch (Cambridge) Bayesian analysis of polarisation data LOFAR MKSP Workshop 4 / 15


Radio sky simulations
Faraday screens

Recipe:
· Halo masses and distribution: N-body simulations, mass functions

and linear theor y
· Scaling relations: Match halo proper ties (B0 , ne0 ) to halo mass · Profiles: Electron gas density (hydro-static equilibrium), B -field

RM sky:

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

5 / 15


Radio sky simulations
Faraday screens

-8000

-4000

0

4000

8000

-1000

-500

0

500

1000

1e+07

amplitude (arbitrary units)

1e+06

100000

10000

10

100

spatial scale (arbitrary units)

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

6 / 15


Bayesian Analysis
Theor y

Bayes' theorem:
posterior
likelihood prior

p(|D , H ) =

p(D |, H ) p(|H ) p (D |H )
evidence

Explore the posterior by Monte Carlo sampling (e.g. Markov Chain Monte Carlo) to obtain best model parameters and confidence inter vals.

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

7 / 15


Polarisation Data Analysis
Parameter constraints: RM inference

MCMC sampling

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

8 / 15


Model selection
Methods

Aim (Ockham's razor):
· Select the model which provides generally the best

description of the data
· Penalise models which are worse or even unsatisfactor y

descriptions

Model selection statistics:
· Akaike Information Criterion · Bayesian Information Criterion · Bayesian Evidence (Jeffreys Information Criterion):

E=

L() ()d

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

9 / 15


Model selection
Methods

Aim (Ockham's razor):
· Select the model which provides generally the best

description of the data
· Penalise models which are worse or even unsatisfactor y

descriptions

Evidence ratios and model selection:
evidence ratio

p(D |H1 ) p(H1 |D ) = p(H2 |D ) p(D |H2 )
model ratio

p(H1 ) p(H2 )
prior ratio

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

9 / 15


Model selection
Methods

Aim (Ockham's razor):
· Select the model which provides generally the best

description of the data
· Penalise models which are worse or even unsatisfactor y

descriptions

Jeffreys Information Criterion:
[ln(E )] < 1 1 < [ln(E )] 2.5 < [ln(E [ln(E )] > 5 Not decisive. < 2.5 Significant. )] < 5 Strong indication. Decisive.

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

9 / 15


Bayesian Analysis
Evidence evaluation

Methods:
· Brute-force attempts (only @ low dimensionality, simple

likelihoods)
· Simulated annealing (during burn-in phase of MCMC sampler) · Nested sampling · ...

More about nested sampling:
· Efficient Monte Carlo evaluation of the evidence. · Evidence integral re-parametrisation:

dX = ()d X () =

L()>

()d
1

· Define X (prior mass) so that E = 0 LdX is uniquely specified · Evaluate Lj = L(Xj ) with 0 < Xn < · · · < X1 < 1

E

n j =1

Lj wj with wj = (Xj

-1

-X

j +1

)/2
LOFAR MKSP Workshop 10 / 15

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data


Bayesian Analysis
Nested sampling

Algorithm:
1 Initialise: j = 0, X0 = 1 and E = 0 2 Get N `live' points and their likelihoods (uniformly) inside the initial prior space 3 j =j +1 4 find lowest Li and remove it from `live' point list 5 Increment evidence as E = E + Li (Xi -1 - Xi +1 )/2 6 Reduce prior volume Xi /Xi p(t ) = Nt N -1
-1

= ti with

7 Replace removed point with new point from with L > Li 8 If (Lmax Xj < E ) then P (E = E + N 1 L(i )/N and i= stop) else (go to (3))

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

11 / 15


Polarisation Data Analysis
Applications

Model Basic model Basic model

parameters & priors RMscr , 0 src , src , psrc (uniform, symmetric) RMscr , 0 src , src , psrc (uniform, `asymmetric', excluding) RMscr , 0 src , src (set), psrc (uniform, symmetric) RMscr , 0 src , src (set), psrc (uniform, symmetric, different ranges)

[ln(E )] 0 -10.44

Basic model (only 3 parameters) Basic model (only 3 parameters)

0.17

1.3

Data: simulated SKA mid band observation
J. Geisbuesch (Cambridge) Bayesian analysis of polarisation data LOFAR MKSP Workshop 12 / 15


Polarisation Data Analysis
Model selection

Model MultiSlab UniSlab
0.1

parameters & priors 10 parameters half the number of parameters

[ln(E )] 0 -118

0.05

Polarised flux [unnormalised]

0

-0.05

-0.1

-0.15

-0.2 9e+08

9.5e+08

1e+09 1.05e+09 1.1e+09 1.15e+09 1.2e+09 1.25e+09 1.3e+09 1.35e+09 1.4e+09 Frequency [Hz]

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

13 / 15


Polarisation Data Analysis
`LOFAR analysis'

Model MultiSlab (bright) UniSlab Model MultiSlab (medium) UniSlab Model MultiSlab (faint) UniSlab Model MultiSlab (ver y faint) UniSlab
J. Geisbuesch (Cambridge)

parameters & priors 10 parameters half the number of parameters parameters & priors 10 parameters half the number of parameters parameters & priors 10 parameters half the number of parameters parameters & priors 10 parameters half the number of parameters
Bayesian analysis of polarisation data

[ln(E )] 0 -46000 [ln(E )] 0 -460 [ln(E )] 0 -6.893 [ln(E )] 0 -0.38
14 / 15

LOFAR MKSP Workshop


Conclusions

We have developed radio polarisation and rotation measure simulations of a high degree of realism · Bayesian analysis and model selection techniques provide efficient and powerful tools to analyse polarisation and RM data · Working in frequency space rather than in RM space can save computing time; the same is true for working in the u-v plane rather than in the image plane
·

J. Geisbuesch (Cambridge)

Bayesian analysis of polarisation data

LOFAR MKSP Workshop

15 / 15