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 = 5 \times 10 ^ { -3 } .
Considering multipoles 32 \leq \ell \leq 121 , we are able to recover estimates for r .
With LiteBIRD only observations , and using the complete pooling model , we recover r = ( 10 \pm 0.6 ) \times 10 ^ { -3 } .
For C-BASS and LiteBIRD observations we find r = ( 7.0 \pm 0.6 ) \times 10 ^ { -3 } using the complete pooling model , and r = ( 5.0 \pm 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 \sim 10 ^ { 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 .