[1103.3505] Bias in lowmultipole CMB reconstructions
Authors:  Craig J. Copi, Dragan Huterer, Dominik J. Schwarz, Glenn D. Starkman 
Abstract:  The largeangle, low multipole cosmic microwave background (CMB) provides a unique view of the largest angular scales in the Universe. Study of these scales is hampered by the facts that we have only one Universe to observe, only a few independent samples of the underlying statistical distribution of these modes, and an incomplete sky to observe due to the interposing Galaxy. Techniques for reconstructing a full sky from partial sky data are well known and have been applied to the large angular scales. In this work we critically study the reconstruction process and show that, in practise, the reconstruction is biased due to leakage of information from the region obscured by foregrounds to the region used for the reconstruction. We conclude that, despite being suboptimal in a technical sense, using the unobscured region without reconstructing is the most robust measure of the true CMB sky. 
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 Posts: 119
 Joined: March 02 2005
 Affiliation: University of Helsinki
[1103.3505] Bias in lowmultipole CMB reconstructions
The most conspicuous anomalies in the CMB are the deviations from statistical homogeneity and isotropy on large angles. There has been debate about their significance and the best method for evaluating them. (See http://cosmocoffee.info/viewtopic.php?t=1582. and http://cosmocoffee.info/viewtopic.php?t=1601&highlight= ).
In particular, the masked CMB sky shows almost zero large angle correlation function, at a level which is here quoted as 99.975% unlikely. However, if you do the analysis on the ILC map, the unlikeliness is only at the 95% level, which is not significant. It seems to me that this result alone shows that something funny is going on: the regions inside and outside the mask seem to behave rather differently!
The present paper argues that the procedures for reconstructing the full sky from a masked sky in fact involve assumptions about what lies behind the mask to such an extent that the reconstruction is significantly biased by the contents of the cut region. The authors therefore conclude (in agreement with their previous work) that the largeangle properties should be studied on the cut sky, not on a reconstructed full sky.
In particular, the masked CMB sky shows almost zero large angle correlation function, at a level which is here quoted as 99.975% unlikely. However, if you do the analysis on the ILC map, the unlikeliness is only at the 95% level, which is not significant. It seems to me that this result alone shows that something funny is going on: the regions inside and outside the mask seem to behave rather differently!
The present paper argues that the procedures for reconstructing the full sky from a masked sky in fact involve assumptions about what lies behind the mask to such an extent that the reconstruction is significantly biased by the contents of the cut region. The authors therefore conclude (in agreement with their previous work) that the largeangle properties should be studied on the cut sky, not on a reconstructed full sky.

 Posts: 54
 Joined: September 28 2004
 Affiliation: University College London
[1103.3505] Bias in lowmultipole CMB reconstructions
You'd probably be interested in http://arxiv.org/abs/1107.5466  it shows how to get rid of this "smoothing bias" in alm reconstruction, proposes a selfconsistent criterion by which to choose a particular estimator for a given reconstruction problem, and shows that the related Quadratic Maximum Likelihood estimator for the angular power spectrum is not affected by this smoothing bias.

 Posts: 119
 Joined: March 02 2005
 Affiliation: University of Helsinki
[1103.3505] Bias in lowmultipole CMB reconstructions
Thanks, I'll have a look.

 Posts: 119
 Joined: March 02 2005
 Affiliation: University of Helsinki
[1103.3505] Bias in lowmultipole CMB reconstructions
It seems to me that the theorydependence of the method in your paper provides support for the statement "the procedures for reconstructing the full sky from a masked sky in fact involve assumptions about what lies behind the mask to such an extent that the reconstruction is significantly biased by the contents of the cut region".

 Posts: 54
 Joined: September 28 2004
 Affiliation: University College London
[1103.3505] Bias in lowmultipole CMB reconstructions
Hi, sorry I did not see this before. The theory dependence is just a consequence of Bayes' theorem. This statement: "the reconstruction is significantly biased by the contents of the cut region" is ambiguous; you haven't explained what you mean by biased.
I pointed out our paper as a detailed analysis of the "smoothing bias" discussed by the Copi et al paper, not as a response to the statement above. The smoothing bias does affect the standard technique for MLextraction of a_lm s, but it is quite easily mitigated by changing the way you smooth. However, the S1/2 statistic discussed by Copi in the context of "what's behind the Galactic plane" is based on C_l extraction, not a_lm extraction. The standard QML estimator is not affected by this smoothing bias.
I pointed out our paper as a detailed analysis of the "smoothing bias" discussed by the Copi et al paper, not as a response to the statement above. The smoothing bias does affect the standard technique for MLextraction of a_lm s, but it is quite easily mitigated by changing the way you smooth. However, the S1/2 statistic discussed by Copi in the context of "what's behind the Galactic plane" is based on C_l extraction, not a_lm extraction. The standard QML estimator is not affected by this smoothing bias.