Cosmological parameter constraints as derived from the Wilkinson Microwave Anisotropy Probe data via Gibbs sampling and the Blackwell-Rao estimator

We study the Blackwell-Rao (BR) estimator of the probability distribution of the angular power spectrum, $P(C_{\ell }\mid \mathbf{d})$, 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 $\ell$, at least at low $\ell$. Our existing sample chains are only long enough to achieve convergence at $\ell \lesssim 40$. In comparison with $P(C_{\ell }\mid \mathbf{d})$ as reported by the WMAP team we find significant differences at these low $\ell$ values. These differences lead to up to $\sim 0.5\sigma$ shifts in the estimates of parameters in a 7-parameter $\Lambda \mathrm{CDM}$ model with nonzero $d{n}_s/d\ln k$. Fixing $d{n}_s/d\ln k=0$ 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 $\ell$ and provide some solutions.