Bayesian parameter estimation in cosmology with Population Monte Carlo

By Darell Moodley (UKZN)Supervisor: Prof. K Moodley (UKZN)

SKA Postgraduate conference, 29 Nov 2010

Applying Bayesian statistics to cosmology

Estimate cosmological parameters for specified models efficiently.

To quantitatively discriminate one model from another in light of data (model testing).

Other relevant applications of parameter estimation include optimizing experimental configurations e.g. MeerKAT antenna parameters.

Bayesian Inference Provides an expression for the

posterior probability that contains the uncertainty regarding parameters of interest.

Difficult to evaluate posterior, due to the normalising constant.

Solution: Use a simulation to draw samples from this distribution.

)()|( xpriorxdatalikelihoodx

Population Monte Carlo (PMC)

• Is an adaptive version of importance sampling.

• Constructs a sequence of samples to provide improved estimations of parameters.

• Based on the fundamental identity:

• are drawn from q, we estimate by,•

dxxqxqxxfdxxxffE )()()()()()(][

Nxx ,...,1

N

n n

nn xq

xxfN

fE1 )(

)()(1][

Methodology

• Draw samples from importance function: , where

and are D component weights that are proportions of the sample taken from each mixture density, , with parameters, .

• Allocate weights to samples: )()(

n

nn xq

xw

D

ddd xxq

1

),()( Nxx ,...,1

q

D ,...,1

Updating rule

Mixture densities

Mixture densities with iterationThe sum of the mixture densities iteratively approaches the target distribution.

Convergence of the importance function

Convergence is reached when the importance function adequately resembles that of the target distribution.

MCMC versus PMC Both methods generate samples that are

representative of complex distributions. MCMC draws from a proposal distribution while

PMC draws from an importance function, that can be chosen to be a mixture of densities.

PMC can reduce computational time and since chains are not correlated, there is no ‘burn-in’ period.

PMC, like MCMC, also has the ability to be parallelisable, hence computationally feasible.

Each iteration produces an independent sample, therefore it can be stopped at any time.

Illustrative exampleBanana shaped distribution that we wish to simulate draws from

Updated importance function after 11 iterations

Wraith et al. 2009

Applications Optimisation

• Search regions of high likelihood to determine optimal parameters. Determine maximum likelihood estimates.

Model selection• Ability to compute Bayesian evidence from

existing chains, hence compute Bayes’ factor for different models.

• Evidence is immediately accessible from the sample used for parameter estimation.

Model Testing

Kilbinger et al. 2010

• PMC can be used to determine the Bayes’ Factor, used to discriminate between models

Test extensions of the standard model with dark-energy and curvature scenarios

Project Objectives Do a systematic study of the PMC

method. Examine the behaviour between

algorithm dependencies and efficiency A quantitative comparison between PMC

and MCMC efficiency Estimation of cosmological parameters

using current data As well as discriminate between

cosmological models.

References

THE ENDTHANK YOU

Adaptive Importance sampling

• Use the Kullback-Liebler distance measure

• Incorporate mixture densities

• D component weights• ,such that

Estimator for the Evidence

• Using importance sampling:• where are the importance weights

for importance distribution q.• Variance is given by

• Want to choose optimal q such that σ is minimised.

N

nnwN

E1

1

)()(

n

nn xq

xw

1)(

)()(22

2 xdxqx

NE

Diagnostics

• Want to maximise so we use the perplexity as an estimate

Application to cosmology

• Compare the cosmological constant and flat ΛCDM model to Dark Matter models

• A Flat (Ωk=0) and Curved (Ωk≠0) model is assumed for each Dark matter model.

Priors for dark energy and curvature models

Specifying the PMC parameters

• For the Dark energy models:• T=10, but can increase if perplexity is still low.• N=7 500• D=10• N/D should be chosen not too small to ensure

numerically stable updating of the component.

Results

• Standard ΛCDM model is favoured.

Testing Stability

• Repeat the PMC runs 25 times.

Primordial fluctuation models

• Dark matter density fluctuations are given by the power spectrum

• with tensor modes

• Parametrise the parameters in terms of the slow-roll parameters.

Assumptions and results

Priors Results

Constraining parameters and Model discrimination

PMC can be used to determine the Bayes’ Factor, used to discriminate between models

PMC is also used to constrain the dark energy equation of state using various data.

Kilbinger et al. 2010

Jeffreys’ scale