Global, Exact Cosmic Microwave Background Data Analysis Using Gibbs Sampling

By Benjamin Dan Wandelt, D. Larson, A. Lakshminarayanan. 2003

177 citations recorded by the source corpus.

We describe an efficient and exact method that enables global Bayesian analysis of cosmic microwave background (CMB) data. The method reveals the joint posterior density (or likelihood for flat priors) of the power spectrum ${C}_{\ensuremath{\ell}}$ and the CMB signal. Foregrounds and instrumental parameters can be simultaneously inferred from the data. The method allows the specification of a wide range of foreground priors. We explicitly show how to propagate the non-Gaussian dependency structure of the ${C}_{\ensuremath{\ell}}$ posterior through to the posterior density of the parameters. If desired, the analysis can be coupled to theoretical (cosmological) priors and can yield the posterior density of cosmological parameter estimates directly from the time-ordered data. The method does not hinge on special assumptions about the survey geometry or noise properties, etc., It is based on a Monte Carlo approach and hence parallelizes trivially. No trace or determinant evaluations are necessary. The feasibility of this approach rests on the ability to solve the systems of linear equations which arise. These are of the same size and computational complexity as the map-making equations. We describe a preconditioned conjugate gradient technique that solves this problem and demonstrate in a numerical example that the computational time required for each Monte Carlo sample scales as ${n}_{p}^{3/2}$ with the number of pixels ${n}_{p}$. We use our method to analyze the data from the Differential Microwave Radiometer on the Cosmic Background Explorer and explore the non-Gaussian joint posterior density of the ${C}_{\ensuremath{\ell}}$ from the Differential Microwave Radiometer on the Cosmic Background Explorer in several projections.

Subfields: Astronomy and Astrophysics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/91235dd69563982d745a3f8e80ac278cbbad5098

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Citations in the corpus

SEVEN-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) OBSERVATIONS: COSMOLOGICAL INTERPRETATION (2010, methodology): “…et al. 2010) to measure the low-l TT power spectrum in l = 2 − 32, and calculate the likelihood using the Gibbs sampling and Blackwell-Rao (BR) estimator (Jewell et al. 2004; Wandelt 2003; Wandelt et al. 2004; O’Dwyer et al. 2004; Eriksen et al. 2004, 2007a,b; Chu et al. 2005; Larson et al. 2007).”
FIVE-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE OBSERVATIONS: COSMOLOGICAL INTERPRETATION (2008, unknown): “We use the Gibbs sampling technique and the Blackwell-Rao (BR) estimator to evaluate the likelihood of the temperature power spectrum at l ≤ 32 (Jewell et al. 2004; Wandelt 2003; Wandelt et al. 2004; O’Dwyer et al. 2004; Eriksen et al. 2004, 2007c,b; Chu et al. 2005; Larson et al. 2007).”
SEVEN-YEAR WILKINSON MICROWAVE ANISOTROPY PROBE (WMAP) OBSERVATIONS: POWER SPECTRA AND WMAP-DERIVED PARAMETERS (2010, methodology): “Temperature (TT) Spectrum For l ≤ 32, the spectrum is obtained using a BlackwellRao estimator applied to a chain of Gibbs samples (Wandelt et al. 2004; Jewell et al. 2004; Chu et al. 2005; Dunkley et al. 2009) based on the 7-year ILC map and the KQ85y7 mask.”
Bayesian physical reconstruction of initial conditions from large-scale structure surveys (2012, background): “…power spectrum from an observed sample of tracers given a survey mask and selection function (see e.g. Feldman et al. 1994; Tegmark & et al. 2004; Wandelt et al. 2004; Eriksen et al. 2004; Percival 2005; Jasche et al. 2010; Elsner & Wandelt 2012); and the much more difficult theoretical problem…”
The Atacama Cosmology Telescope: likelihood for small-scale CMB data (2013, methodology): “…distribution has a known form, and has been used, for example, to estimate the large-scale CMB power spectrum, and to marginalize over Galactic foregrounds (e.g., Wandelt et al. 2004; Jewell et al. 2004; Eriksen et al. 2004; Dunkley et al. 2009; Larson et al. 2011) ), is then also a Gaussian.”
Planck intermediate results. LVII. Joint Planck LFI and HFI data processing (2020, methodology): “Our approach to this problem was described in general form by Jewell et al. (2004), Wandelt et al. (2004), and Eriksen et al. (2004), and later implemented by Hinshaw et al. (2009) for WMAP.”