I agree likelihoods should be optimal, but I don't think the algorithm in itself is doing any E/B mode separation (all full sky noise free maps will have some non-zero B mode everywhere, including in the observed region). To separate the modes you'd need to project the full sky sampled Elm and Blm into the linear combinations of the full sky E and B modes that are pure B and pure E

I don't see why we would need to project the full sky sampled Elm and Blm into pure

B and pure

E modes outside the mask. Working with pure

E and

B modes outside the mask is a useful technique if you don't already have the Elm and Blm coefficients for the full sky. However, we already do have those coefficients, so we can go directly from there to estimates of the power spectra: [tex]C_\ell^{EE}[/tex] and [tex]C_\ell^{BB}[/tex] (and [tex]C_\ell^{EB}[/tex], if you like).

The simplest way to do that would be to use formulas like

[tex]\hat{C}_\ell^{EE} = \frac{1}{2\ell+1}\sum_m |a^{E}_{\ell m}|^2[/tex]

I guess I have been assuming we are primarily interested in the power spectrum. If we want to know what the actual

B modes are, perhaps to map out the weak lensing potential, then Gibbs sampling still samples realizations of the

B modes. (We would, however, have to modify our assumption of isotropic Gaussianity for the

B modes, since weak lensing doesn't satisfy that.)

It is true that our

E-

B separation happens after the fact. All the Gibbs sampling code does is sample full-sky polarized realizations according to their likelihood. However, our point is that when you have a full-sky realizations of the polarization, then breaking that up into the

E and

B parts is trivial (now that the HEALPix code has been written, anyway).

(all full sky noise free maps will have some non-zero B mode everywhere, including in the observed region)

I don't understand this statement. It's possible to determine the "curl" of a polarization field with a differential operator on the sphere (act with edthbar twice and take the imaginary part). This curl operator determines how much

B mode there is at a given location. If the Gibbs sampler were run on a zero-noise masked sky that contained no

B modes at all, then I believe the resulting samples would only have "curl" or

B-ness inside the mask, not everywhere.

Let me know if I've misunderstood what you said.