the pseudo-$C_{l}$s {#sec:inverse}
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For the purpose of parameter estimation, it is not strictly necessary to attempt to invert the pseudo-$C_l$s to (unbiased) power spectrum estimates. Forming the pseudo-$C_l$s can be regarded as a mildly-lossy form of data compression of the maps of Stokes parameters, and inferences about cosmological models can be made directly from them if their covariance – or, better, their full sampling distribution – can be calculated [@wandelt01b]. The calculation of the signal contribution to these covariances is the main topic of this paper. However, for presentation purposes it is useful to be able to plot quantities that are simply related to the underlying power spectra. At the least this requires some approximate deconvolution of the geometric effects of the weight function from the pseudo-$C_l$s.

For observations covering enough of the sky[^3], the matrices $P_{ll'}$ and $M_{ll'}$ will be invertible. In this case we easily obtain unbiased estimates of $C_l^E$ and $C_l^B$ from $$\hat{C}_l^E + \hat{C}_l^B = \sum_{l'} (P+M)^{-1}_{ll'}(\tilde{C}_l^E +
\tilde{C}_l^B), \quad
\hat{C}_l^E - \hat{C}_l^B = \sum_{l'} (P-M)^{-1}_{ll'}(\tilde{C}_l^E -
\tilde{C}_l^B).
\label{eq:16}$$ Alternatively, we can form unbiased estimates of the correlation functions of the Stokes parameters from the polarization pseudo-$C_l$s (or directly from the maps), and then invert 