[astro-ph/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 all-sky 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 auto-power 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 full-sky coverage. Simple extensions of auto-power 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 all-sky survey.
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Antony Lewis
Posts: 1659
Joined: September 23 2004
Affiliation: University of Sussex

[astro-ph/0604547] Likelihood methods for the combined analy

Post by Antony Lewis » April 27 2006

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 astro-ph/0511703. I did spend a little time myself trying to generalise this to polarisation, but came to the conclusion that it was non-trivial.

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 near-full sky observations perhaps it would be more accurate just to use the full 'Wishart' distribution with some effective degrees of freedom, neglecting (small) off-l correlations? (obviously with exact likelihood on large scales)

Michael Brown
Posts: 4
Joined: April 27 2006
Affiliation: University of Manchester

[astro-ph/0604547] Likelihood methods for the combined analy

Post by Michael Brown » April 27 2006

Hi Antony,

Thanks - we have now read astro-ph/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 auto-power 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 non-Gaussian 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 all-sky likelihood (which we showed is given by a Wishart distribution) well in the directions of the cross- and auto-power spectra.


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