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● 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
● 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
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