Properties and use of CMB power spectrum likelihoods
  Fast robust methods for calculating likelihoods from CMB 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
approximation, 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 CMB 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-ell likelihood with a more exact low-ell harmonic-space likelihood as a
practical method for accurate likelihood calculation on all scales.
