Properties and use of CMB power spectrum likelihoods

Fast robust methods for calculating likelihoods from cosmic microwave background observations on small scales generally rely on approximations based on a set of power spectrum estimators and their covariances. We investigate the optimality of these approximations, how accurate the covariance needs to be, and how to estimate the covariance from simulations. For a simple case with azimuthal symmetry we compare optimality of hybrid pseudo-C-l cosmic microwave background power spectrum estimators with the exact result, indicating that the loss of information is not negligible, but neither is it enough to have a large effect on standard parameter constraints. We then discuss the number of samples required to estimate the covariance from simulations, with and without a good analytic approximation, and assess the use of shrinkage estimators. Finally we discuss how to combine an approximate high-l likelihood with a more exact low-l harmonic-space likelihood as a practical method for accurate likelihood calculation on all scales.