  We present a Gibbs sampling solution to the map-making problem for CMB
measurements, building on existing destriping methodology. Gibbs sampling
breaks the computationally heavy destriping problem into two separate steps;
noise filtering and map binning. Considered as two separate steps, both are
computationally much cheaper than solving the combined problem. This provides a
huge performance benefit as compared to traditional methods, and allows us for
the first time to bring the destriping baseline length to a single sample. We
apply the Gibbs procedure to simulated Planck 30 GHz data. We find that gaps in
the time-ordered data are handled efficiently by filling them with simulated
noise as part of the Gibbs process. The Gibbs procedure yields a chain of map
samples, from which we may compute the posterior mean as a best-estimate map.
The variation in the chain provides information on the correlated residual
noise, without need to construct a full noise covariance matrix. However, if
only a single maximum-likelihood frequency map estimate is required, we find
that traditional conjugate gradient solvers converge much faster than a Gibbs
sampler in terms of total number of iterations. The conceptual advantages of
the Gibbs sampling approach lies in statistically well-defined error
propagation and systematic error correction, and this methodology forms the
conceptual basis for the map-making algorithm employed in the BeyondPlanck
framework, which implements the first end-to-end Bayesian analysis pipeline for
CMB observations.
