Antony Lewis wrote:* Compressing the data into power spectrum estimators, as WMAP do at high

l, should be suboptimal but unbiased as long as a valid likelihood approximation is used. I'm therefore surprised to see apparent shifts in parameters rather than changes in the error bar. When testing with simulations, I've found that the WMAP-like likelihood approximations work just fine. Even evident deviations from the assumed likelihood model has almost no effect because the errors tend to cancel between

l (see

0804.3865). Indeed just using a pure-Gaussian likelihood approximation works fine in almost all realisations. I'm sure WMAP have also extensively tested their method for biases in simulations. I wonder if the authors reproduce such shifts in idealized simulations?

No, we haven't compared to simulations. Instead, we have checked that the approximation matches the exact likelihood, which of course is a an even better approach – not only is it statistically unbiased (which is all you can check with MC simulations), but it gives the right answer in each particular realization. The loophole, of course, is the point that the validation was done at low resolution, while the WMAP5 analysis is at high resolution. But it's reasonable to assume that if it works at low l's, it works at high l's, I think, since there are less correlations there, and the distributions are intrinsically more Gaussian. But there's a loophole there, yes.

Still, the observed shift is somewhat surprising, yes, but no more so than it was when we saw the same thing in WMAP3: In that case, we found a shift of 0.4 sigma in

n_{s} when increasing lmax from 12 to 30 for the exact part. Then, at first we thought the shift was due to diffuse foregrounds and processing errors, and it took some time before Eiichiro Komatsu and we figured out that it was actually the likelihood approximation that caused this: Switching from the approximate MASTER-likelihood to the exact likelihood between

l=12 and 30 increased

n_{s} by 0.4 sigma. And this is why the WMAP team adopted lmax=30 in their 5-year analysis.

And now we find a very similar effect by increasing lmax from 30 to 200...

Also, note that statistical unbiasedness does not mean "the same as the exact likelihood answer in every realization".

Antony Lewis wrote:
* This being the case, could the shifts be due to something else, e.g. differences in beam modelling? The paper doesn't comment on what they do about the beams at low l, even though the beam transfer function is not unity even at l<200.

Note that we do take into account the actual DA specific beams for each map (V1 and V2). The analysis is done at full WMAP resolution of Nside=512 (not a degraded version of them), so this is straightforward to do in the Gibbs sampler.

Antony Lewis wrote:
* In the conclusions they comment that their approach allows seamless propagation of systematic effects such as beam errors. I'm not convinced by this: a beam error essentially shifts the entire spectrum up and down. The approximation used in the paper fits the maginalized C_{l} distributions at each l separately, which in the case of beam errors are actually strongly correlated between l. Is there any reason to expect this to work?

It's at least as likely as for the WMAP approach, which also does this quadratically. Note that the assumption in the GBR estimator is that only 2-point correlations in

l are important, not 3-point. Essentially, what beam uncertainties mean is that there will be large-scale correlations in the correlation matrix, and we can either compute these by MC (as in the Gibbs sampler), or by converting the known beam covariance matrix to Gaussianized

x-space and then add it to

C. Both should work quite well, I think, but there is of course always a question of convergence, and needs to be tested. But I think there's good reasons to expect that it will work, yes – and it should at least work better than the current WMAP approach.