Application of Monte Carlo algorithms to the Bayesian analysis of the cosmic microwave background

Power spectrum estimation and evaluation of associated errors in the presence of incomplete sky coverage; nonhomogeneous, correlated instrumental noise; and foreground emission are problems of central importance for the extraction of cosmological information from the cosmic microwave background (CMB). We develop a Monte Carlo approach for the maximum likelihood estimation of the power spectrum. The method is based on an identity for the Bayesian posterior as a marginalization over unknowns, and maximization of the posterior involves the computation of expectation values as a sample average from maps of the cosmic microwave background and foregrounds given some current estimate of the power spectrum or cosmological model, as well as some assumed statistical characterization of the foregrounds. Maps of the CMB and foregrounds are sampled by a linear transform of a Gaussian white-noise process, implemented numerically with conjugate gradient descent. For time series data with N-t samples and N pixels on the sphere, the method has a computational expense KO(N-2 + N-t log N-t), where K is a prefactor determined by the convergence rate of conjugate gradient descent. Preconditioners for conjugate gradient descent are given for scans close to great circle paths, and the method allows partial sky coverage for these cases by numerically marginalizing over the unobserved, or removed, region.