Application of Monte Carlo Algorithms to the Bayesian Analysis of the Cosmic Microwave Background

Power spectrum estimation and evaluation of associated errors in the presenceof incomplete sky coverage; non-homogeneous, correlated instrumental noise; andforeground emission is a problem of central importance for the extraction ofcosmological information from the cosmic microwave background. We develop a Monte Carlo approach for the maximum likelihood estimation of the powerspectrum. The method is based on an identity for the Bayesian posterior as amarginalization over unknowns. Maximization of the posterior involves thecomputation of expectation values as a sample average from maps of the cosmicmicrowave background and foregrounds given some current estimate of the powerspectrum or cosmological model, and some assumed statistical characterizationof the foregrounds. Maps of the CMB are sampled by a linear transform of aGaussian white noise process, implemented numerically with conjugate gradientdescent. 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} +AFw-log N_{t}],where K is a prefactor determined by the convergence rate of conjugate gradientdescent. Preconditioners for conjugate gradient descent are given for scansclose to great circle paths, and the method allows partial sky coverage forthese cases by numerically marginalizing over the unobserved, or removed,region.Comment: submitted to Ap