Hierarchical Bayesian CMB component separation with the No-U-Turn Sampler

Abstract

In this paper, we present a novel implementation of Bayesian cosmic microwave background (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 pixel spectral parameters as being drawn from underlying hyperdistributions. We validate the algorithm against simulations of the LiteBIRD and C-Band All-Sky Survey (C-BASS) experiments, with an input tensor-to-scalar ratio of r = 5 × 10−3. Considering multipoles 30 ≤ ℓ < 180, we are able to recover estimates for r. With LiteBIRD-only observations, and using the complete pooling model, we recover r = (12.9 ± 1.4) × 10−3. For C-BASS and LiteBIRD observations we find r = (9.0 ± 1.1) × 10−3 using the complete pooling model, and r = (5.2 ± 1.0) × 10−3 using the hierarchical model. Unlike the complete pooling model, the hierarchical model captures pixel-scale spatial variations in the foreground spectral parameters, and therefore produces cosmological parameter estimates with reduced bias, without inflating their uncertainties. Measured by the rate of effective sample generation, NUTS offers performance improvements of ∼103 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.

Improved Galactic Foreground Removal for B-Modes Detection with Clustering Methods

“…An obvious area of improvement is to optimize the definition of sky regions. The characteristics of clusters obtained in Grumitt et al (2020) were almost homogeneous across the sky: showing approximately the same size and similar (convex) shapes. However, the mean-shift cluster size is highly dependent on the choice of the bandwidth parameter in the definition of the kernels to identify the clusters.…”

“…7, results into shifting to higher multipoles the contribution due to this partition noise. Grumitt et al (2020) proposed a hierarchical approach to overcome exactly this limitation and we plan to integrate it in a future work.…”

“…Recently, several techniques to encompass the spatial variability of the foregrounds have been proposed by decomposing the map into wavelets or needlets (Basak & Delabrouille 2011;Remazeilles et al 2018;Wagner-Carena et al 2019;Irfan et al 2019), by partitioning the sky into regions (clusters) according to the similarity of foreground properties and their location in the sky (Khatri 2019;Grumitt et al 2020).…”

“…The approach is similar to the one shown in Grumitt et al (2020) but we choose a different clustering method to optimally divide the sky. In fact, Grumitt et al (2020) used the mean-shift algorithm (Krzanowski & Lai 1988) to locate overdensities in the context of image segmentation.…”

“…The approach is similar to the one shown in Grumitt et al (2020) but we choose a different clustering method to optimally divide the sky. In fact, Grumitt et al (2020) used the mean-shift algorithm (Krzanowski & Lai 1988) to locate overdensities in the context of image segmentation. They developed a parametric bayesian component separation algorithm that makes use of a clustering analysis to forecast the accuracy of component separation for a LiteBIRD-like CMB space satellite mission (Sugai et al 2020).…”