MCMC of the CMB power spectrum, and

convergence

Zubair and JenCosmology 321

● Random samples of parameter space

● Use CAMB for numerical calculation of Cl

● Calculate likelihood using WMAP 7-yr results

● Accept or reject using Metropolis Hastings

algorithm

– tuning

MCMC of CMB power spectrum

MCMC of CMB power spectrum

● Results

– Ω_cdmh2= 0.103 ± .0032

– Ω_baryonsh2 = .0214 ± .0004

– w= -1.049 ± .028

– H0= 72.16 ± 8.6 km s^-1 Mpc^-1

● WMAP 7-yr (Komatsu et al 2010)

– Ω_cdmh2= 0.112 ± .0035

– Ω_baryonsh2 = .026 ± .00053

– w = -.98 ± .053

– H0 = 70.4 ± 1.4 km s^-1 Mpc^-1

Convergence

● MCMC algorithm approached stationary distribution

● When have we converged?– Stop burn-in

● How many steps to produce representative sample of posterior distribution?

– Safe to stop sampling

Test of convergence

● Theoretical – predetermine number of iterations given a model

– Involved, sophisticated mathematics

– Tends to give unreasonably high number of iterations

● Diagnostic tools– Applied to the output of algorithm

Tests of Convergence

● History plot of parameter vs. iterations– Should show no trends, and should snake

around the mean

– Does not guarantee escape from a local minima, or sampling entire distribution

Tests of Convergence

● Autocorrelation● Sequential draws are correlated. Should

decrease with the number of lag

omega_matter omega_lambda

Graph by Ken Hanson, Los Alamos

Tests of Convergence

● Running means – if there are enough iterations to describe the posterior distribution, then subsequent draws should not change the mean or the variance.

omega_matter omega_baryons

Tests of Convergence

● Kernel Distribution, histogram– Lumpiness or multimodal

Tests of Convergence

● Gelman and Rubin (1992)– Use overdispersed estimate of distribution to

generate independent starting points

– Compare ~last ½ of multiple chains to produce a quantity with both the variance between chains and within individual chains,

Tests of Convergence

● Gelman Rubin cont.– Slow mixing samples have a large variance

between the means because of starting points

– When the statistic goes to 1, then use combination of last ½ of chains

– Escaped the influence of the starting point, variance of individual chains dominates variance between each chain

– Draw back: need to produce overdispersed distribution; throwing away several computations due to multiple chains

Tests of Convergence

● Gelman Rubin Statistic– 3 chains from salt-Sne data,1000 steps each

– n=1000, m=3, df=3, B=3.98E-5,W=.0022● R^1/2 = 1.73

Tests of convergence

● Geweke (1992)– Single chain test, analogy to spectral analysis

– Essentially a test to see if mean and var. of one half is equal to mean and var. of second half.

– No specified procedure

– Duncley et al 2004

● Multivariate diagnostics

Tests of convergence

Cowels and Carlin

Conclusions

● MCMC powerful tool to probe posterior distribution

● Verify convergence after burn-in with diagnostic tools

● Knowing about target density can be benificial● Careful choice of priors, parameters, and

proposal density increases efficiency

Refrences

● Lewis and Bridle, 2002● Duncley et al, 2004● Hanson and Cunningham, 1998● Cowles and Carlin, MCMC convergence

diagnostics● camb.info, cosmomc,

http://lambda.gsfc.nasa.gov/