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Register a new userHierarchical Bayesian CMB Component Separation with the No-U-Turn Sampler
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Key to any cosmic microwave background (CMB) analysis is the separation of the CMB from foreground contaminants. In this paper we present a novel implementation of Bayesian CMB component separation. We sample from the full posterior distribution using the No-U-Turn Sampler (NUTS), a gradient based sampling algorithm. Alongside this, we introduce new foreground modelling approaches. We use the mean-shift algorithm to define regions on the sky, clustering according to naively estimated foreground spectral parameters. Over these regions we adopt a complete pooling model, where we assume constant spectral parameters, and a hierarchical model, where we model individual spectral parameters as being drawn from underlying hyper-distributions. We validate the algorithm against simulations of the LiteBIRD and C-BASS experiments, with an input tensor-to-scalar ratio of $r=5times 10^{-3}$. Considering multipoles $32leqellleq 121$, we are able to recover estimates for $r$. With LiteBIRD only observations, and using the complete pooling model, we recover $r=(10pm 0.6)times 10^{-3}$. For C-BASS and LiteBIRD observations we find $r=(7.0pm 0.6)times 10^{-3}$ using the complete pooling model, and $r=(5.0pm 0.4)times 10^{-3}$ using the hierarchical model. By adopting the hierarchical model we are able to eliminate biases in our cosmological parameter estimation, and obtain lower uncertainties due to the smaller Galactic emission mask that can be adopted for power spectrum estimation. Measured by the rate of effective sample generation, NUTS offers performance improvements of $sim10^3$ over using Metropolis-Hastings to fit the complete pooling model. The efficiency of NUTS allows us to fit the more sophisticated hierarchical foreground model, that would likely be intractable with non-gradient based sampling algorithms.
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We present a novel technique for Cosmic Microwave Background (CMB) foreground subtraction based on the framework of blind source separation. Inspired by previous work incorporating local variation to Generalized Morphological Component Analysis (GMCA), we introduce Hierarchical GMCA (HGMCA), a Bayesian hierarchical graphical model for source separation. We test our method on $N_{rm side}=256$ simulated sky maps that include dust, synchrotron, free-free and anomalous microwave emission, and show that HGMCA reduces foreground contamination by $25%$ over GMCA in both the regions included and excluded by the Planck UT78 mask, decreases the error in the measurement of the CMB temperature power spectrum to the $0.02-0.03%$ level at $ell>200$ (and $<0.26%$ for all $ell$), and reduces correlation to all the foregrounds. We find equivalent or improved performance when compared to state-of-the-art Internal Linear Combination (ILC)-type algorithms on these simulations, suggesting that HGMCA may be a competitive alternative to foreground separation techniques previously applied to observed CMB data. Additionally, we show that our performance does not suffer when we perturb model parameters or alter the CMB realization, which suggests that our algorithm generalizes well beyond our simplified simulations. Our results open a new avenue for constructing CMB maps through Bayesian hierarchical analysis.
The polarization modes of the cosmological microwave background are an invaluable source of information for cosmology, and a unique window to probe the energy scale of inflation. Extracting such information from microwave surveys requires disentangling between foreground emissions and the cosmological signal, which boils down to solving a component separation problem. Component separation techniques have been widely studied for the recovery of CMB temperature anisotropies but quite rarely for the polarization modes. In this case, most component separation techniques make use of second-order statistics to discriminate between the various components. More recent methods, which rather emphasize on the sparsity of the components in the wavelet domain, have been shown to provide low-foreground, full-sky estimate of the CMB temperature anisotropies. Building on sparsity, the present paper introduces a new component separation technique dubbed PolGMCA (Polarized Generalized Morphological Component Analysis), which refines previous work to specifically tackle the estimation of the polarized CMB maps: i) it benefits from a recently introduced sparsity-based mechanism to cope with partially correlated components, ii) it builds upon estimator aggregation techniques to further yield a better noise contamination/non-Gaussian foreground residual trade-off. The PolGMCA algorithm is evaluated on simulations of full-sky polarized microwave sky simulations using the Planck Sky Model (PSM), which show that the proposed method achieve a precise recovery of the CMB map in polarization with low noise/foreground contamination residuals. It provides improvements with respect to standard methods, especially on the galactic center where estimating the CMB is challenging.
A well-tested and validated Gibbs sampling code, that performs component separation and CMB power spectrum estimation, was applied to the {it WMAP} 5-yr data. Using a simple model consisting of CMB, noise, monopoles and dipoles, a ``per pixel low-frequency power-law (fitting for both amplitude and spectral index), and a thermal dust template with fixed spectral index, we found that the low-$ell$ ($ell < 50$) CMB power spectrum is in good agreement with the published {it WMAP}5 results. Residual monopoles and dipoles were found to be small ($lesssim 3 mu$K) or negligible in the 5-yr data. We comprehensively tested the assumptions that were made about the foregrounds (e.g. dust spectral index, power-law spectral index prior, templates), and found that the CMB power spectrum was insensitive to these choices. We confirm the asymmetry of power between the north and south ecliptic hemispheres, which appears to be robust against foreground modeling. The map of low frequency spectral indices indicates a steeper spectrum on average ($beta=-2.97pm0.21$) relative to those found at low ($sim$GHz) frequencies.
Markov Chain Monte Carlo (MCMC) is a powerful method for drawing samples from non-standard probability distributions and is utilized across many fields and disciplines. Methods such as Metropolis-Adjusted Langevin (MALA) and Hamiltonian Monte Carlo (HMC), which use gradient information to explore the target distribution, are popular variants of MCMC. The Sequential Monte Carlo (SMC) sampler is an alternative sampling method which, unlike MCMC, can readily utilise parallel computing architectures and also has tuning parameters not available to MCMC. One such parameter is the L-kernel which can be used to minimise the variance of the estimates from an SMC sampler. In this letter, we show how the proposal used in the No-U-Turn Sampler (NUTS), an advanced variant of HMC, can be incorporated into an SMC sampler to improve the efficiency of the exploration of the target space. We also show how the SMC sampler can be optimized using both a near-optimal L-kernel and a Hamiltonian proposal
We describe and implement an exact, flexible, and computationally efficient algorithm for joint component separation and CMB power spectrum estimation, building on a Gibbs sampling framework. Two essential new features are 1) conditional sampling of foreground spectral parameters, and 2) joint sampling of all amplitude-type degrees of freedom (e.g., CMB, foreground pixel amplitudes, and global template amplitudes) given spectral parameters. Given a parametric model of the foreground signals, we estimate efficiently and accurately the exact joint foreground-CMB posterior distribution, and therefore all marginal distributions such as the CMB power spectrum or foreground spectral index posteriors. The main limitation of the current implementation is the requirement of identical beam responses at all frequencies, which restricts the analysis to the lowest resolution of a given experiment. We outline a future generalization to multi-resolution observations. To verify the method, we analyse simple models and compare the results to analytical predictions. We then analyze a realistic simulation with properties similar to the 3-yr WMAP data, downgraded to a common resolution of 3 degree FWHM. The results from the actual 3-yr WMAP temperature analysis are presented in a companion Letter.
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