This paper examines various approximations to CMB likelihood functions, and attempts to generalise to polarisation. The conclustion seems to be that accurately getting the shape of the polarisation function from simple fittings is difficult  I agree. In Eq 43 the authors appear to be taking the log of things like C^{TE} that can be negative  so does this not fail to return sensible likelihoods for some models?
There is a nice family of approximations the authors have not discussed: C^{1/3} and C^{1/3} parameterizations are both much better than Gaussian + log normal  see the appendix of astroph/0511703. I did spend a little time myself trying to generalise this to polarisation, but came to the conclusion that it was nontrivial.
Note that WMAP are currently using small scale polarized likelihood functions that are quite inadequate once the data becomes much better (e.g. Planck).
Given the complications, for nearfull sky observations perhaps it would be more accurate just to use the full 'Wishart' distribution with some effective degrees of freedom, neglecting (small) offl correlations? (obviously with exact likelihood on large scales)
[astroph/0604547] Likelihood methods for the combined analysis of CMB temperature and
Authors:  Will J. Percival (ICG, University of Portsmouth), Michael L. Brown (IfA, University of Edinburgh) 
Abstract:  We consider the shape of the likelihood and posterior surfaces to be used when fitting cosmological models to CMB temperature and polarisation power spectra measured from experiments. In the limit of an allsky survey with Gaussian distributed pixel noise we show that the true combined likelihood of the four CMB power spectra (TT, TE, EE & BB) has a Wishart distribution and we discuss the properties of this function. We compare various fits to the posterior surface of the Cls, both in the case of a single autopower spectrum and for a combination of temperature and polarisation data. In the latter case, it is important that the fits reduce to the Wishart distribution in the limit of near fullsky coverage. Simple extensions of autopower spectrum fits to include polarisation data generally fail to match correlations between the different power spectra in this limit. Directly fitting pixel values on large scales, as undertaken by the WMAP team in their analysis of the 3 year data, avoids the complications of characterising the shape of the posterior for the power spectra. Finally we demonstrate the importance of the posterior distribution shape on analytic marginalisation, and provide a formula for analytic marginalisation over a calibration error given an allsky survey. 
[PDF] [PS] [BibTex] [Bookmark] 

 Posts: 1522
 Joined: September 23 2004
 Affiliation: University of Sussex
 Contact:

 Posts: 4
 Joined: April 27 2006
 Affiliation: University of Manchester
[astroph/0604547] Likelihood methods for the combined analy
Hi Antony,
Thanks  we have now read astroph/0511703 and their likelihood fit is very interesting. We'll include this in the next version of the paper.
With the offset lognormal function, it is true that there is the possiblity that you end up trying to take the logarithm of a negative number for certain models. This was true even for an autopower spectrum with a negative "a" parameter (the free parameter in Eq 43) multiplied by the measured power. However, letting [tex]a \, \to \infty[/tex] (for +ve measured [tex]C_l[/tex]) or [tex]a \, \to \infty[/tex] (for ve measure [tex]C_l[/tex]) will make the distribution tend to a Gaussian, and the fit is OK for all models in this limit. In general, we're only concerned with the slightly nonGaussian regime where this taking the log of a negative number isn't a problem (except for extremely wacky models). An alternative way of thinking about this is to say that with this fitting function you can only compare certain models that end up taking a log of a positive number. All other models are assumed to be highly unlikely. This doesn't stop this function being used as a fitting function  in fact, as we show in the paper, it can match the allsky likelihood (which we showed is given by a Wishart distribution) well in the directions of the cross and autopower spectra.
cheers,
Michael.
Thanks  we have now read astroph/0511703 and their likelihood fit is very interesting. We'll include this in the next version of the paper.
With the offset lognormal function, it is true that there is the possiblity that you end up trying to take the logarithm of a negative number for certain models. This was true even for an autopower spectrum with a negative "a" parameter (the free parameter in Eq 43) multiplied by the measured power. However, letting [tex]a \, \to \infty[/tex] (for +ve measured [tex]C_l[/tex]) or [tex]a \, \to \infty[/tex] (for ve measure [tex]C_l[/tex]) will make the distribution tend to a Gaussian, and the fit is OK for all models in this limit. In general, we're only concerned with the slightly nonGaussian regime where this taking the log of a negative number isn't a problem (except for extremely wacky models). An alternative way of thinking about this is to say that with this fitting function you can only compare certain models that end up taking a log of a positive number. All other models are assumed to be highly unlikely. This doesn't stop this function being used as a fitting function  in fact, as we show in the paper, it can match the allsky likelihood (which we showed is given by a Wishart distribution) well in the directions of the cross and autopower spectra.
cheers,
Michael.