Marginal Distributions for Cosmic Variance Limited Cosmic Microwave Background Polarization Data

We provide computationally convenient expressions for all marginal distributions of the polarization cosmic microwave background (CMB) power spectrum distribution P (C |σ ), where C = { CTT , C TE , C EE , C BB } denotes the set of ensemble-averaged polarization CMB power spectra, and σ = { σ TT , σ TE , σ EE , σ BB } the set of the realization-specific polarization CMB power spectra. This distribution describes the CMB posterior power spectrum for cosmic variance limited data. The expressions derived here are general, and may be useful in a wide range of applications. Two specific applications are described in this paper. First, we employ the derived distributions within the CMB Gibbs sampling framework and demonstrate a new conditional CMB power spectrum sampling algorithm that allows for different binning schemes for each power spectrum. This is useful because most CMB experiments have very different signal-to-noise ratios for temperature and polarization. Second, we provide new Blackwell–Rao estimators for each of the marginal polarization distributions, which are relevant to power spectrum and likelihood estimation. Because these estimators represent marginals, they are not affected by the exponential behavior of the corresponding joint expression, but converge quickly.