Amended Gibbs samplers for Cosmic Microwave Background power spectrum estimation

We study different variants of the Gibbs sampler algorithm from the perspective of their applicability to the estimation of power spectra of the cosmic microwave background (CMB) anisotropies. We focus on approaches which aim at reducing the cost of a computationally heavy constrained realization step and capitalize on the interweaving strategy to ensure that our algorithms mix well for high and low signal-to-noise ratio components. In particular, we propose an approach, which we refer to as Centered overrelax, which avoids the constraint realization step completely at the cost of additional, auxiliary variables and the need for overrelaxation. We demonstrate these variants and compare their merits on full and cut sky simulations, quantifying their performance in terms of an effective sample size (ESS) per second. We find that on nearly full-sky, satellite-like data, the proposed Gibbs sampler with overrelaxation performs between one and two orders of magnitude better than the usual Gibbs sampler potentially providing an interesting alternative to the currently favored approaches.