  We study the Blackwell-Rao (BR) estimator of the probability distribution of
the angular power spectrum, P(C_l|d), by applying it to samples of full-sky
no-noise CMB maps generated via Gibbs sampling. We find the estimator, given a
set of samples, to be very fast and also highly accurate, as determined by
tests with simulated data. We also find that the number of samples required for
convergence of the BR estimate rises rapidly with increasing l, at least at low
l. Our existing sample chains are only long enough to achieve convergence at l
less than about 40. In comparison with P(C_l|d) as reported by the WMAP team we
find significant differences at these low l values. These differences lead to
up to approximately 0.5 sigma shifts in the estimates of parameters in a
7-parameter Lambda CDM model with non-zero dn_s/dln(k). Fixing dn_s/dln(k)= 0
makes these shifts much less significant.
  Unlike existing analytic approximations, the BR estimator can be
straightforwardly extended for the case of power spectra from correlated
fields, such as temperature and polarization. We discuss challenges to
extending the procedure to higher l and provide some solutions.
