Hierarchical Bayesian CMB Component Separation with the No-U-Turn Sampler
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 pixel spectral parameters as being drawn from underlying hyper-distributions. We validate the algorithm against simulations of the \textit{LiteBIRD} and C-BASS experiments, with an input tensor-to-scalar ratio of $r=5\times 10^{-3}$. Considering multipoles $30\leq\ell<180$, we are able to recover estimates for $r$. With \textit{LiteBIRD}-only observations, and using the complete pooling model, we recover $r=(12.9\pm 1.4)\times 10^{-3}$. For C-BASS and \textit{LiteBIRD} observations we find $r=(9.0\pm 1.1)\times 10^{-3}$ using the complete pooling model, and $r=(5.2\pm 1.0)\times 10^{-3}$ 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 the their uncertainties. 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.