In this paper we adopt a Bayesian approach to the separation of foreground components from CMBR emission for satellite observations. We find that by assuming a suitable Gaussian prior in Bayes' theorem for the sky emission, we recover the standard Wiener filter (WF) approach. Alternatively, we may assume an entropic prior, based on information-theoretic considerations alone, from which we derive a maximum entropy method (MEM). We apply these two methods to the problem of separating the different physical components of sky emission.
By assuming a Gaussian prior in Bayes' theorem, we derived the standard form of the Wiener filter. This approach is optimal in the sense that it is the linear filter for which the variance of the reconstruction residuals is minimised. This is true both in the Fourier domain and the map domain. Nevertheless, it is straightforward to show that this algorithm leads to maps with power spectra that are biased compared to the true spectra, and this leads us to consider variants of the standard Wiener filter.
At a given value of , the estimator of the azimuthally averaged power spectrum for the th physical component is obtained simply by calculating the average value of over those modes for which , i.e.
where is the number of measured Fourier modes satisfying .
The bias in the power spectrum of the standard WF map reconstruction may be quantified by introducing, for each physical component, a quality factor at each Fourier mode (Bouchet et al. (1997)). This factor is given by
where is the response matrix of the observations at the Fourier mode , as defined in equation (3), and is the corresponding Wiener matrix given in equation (14). The quality factor varies between unity (in the absence of noise) and zero. If is the WF estimate of the th component of the signal vector at and is the actual value, then it is straightforward to show that
Thus, in similar way, the expectation value of the naive power spectrum estimator defined in (28) is given by , where is the average of the quality factors at each Fourier mode satisfying ; thus the estimator in equation (28) is biased and should be replaced by .
It is clearly unsatisfactory, however, to produce reconstructed maps with biased power spectra and, from the above discussion, we might consider using the matrix with elements to perform the reconstructions. Bouchet et al. (1997) shows that this leads to reconstructed maps that do indeed possess unbiased power spectra and, moreover, the method is less sensitive to the assumed input power spectra. However, one finds in this case that the variance of the reconstruction residuals is increase by a factor compared to those obtained with the standard WF and so the reconstructed maps appear somewhat noisier.
Another variant of the Wiener filter technique has been proposed by Tegmark & Efstathiou (1996) and Bouchet et al. (1997), and uses the matrix to perform the reconstructions. This approach has the advantage that the reconstruction of the th physical component is independent of its assumed input power spectrum. Nevertheless, for this technique the variance of the reconstruction residuals is found to be increased by the factor as compared to the standard WF, which results in even noisier reconstructed maps.
As a final variant, Tegmark (1997) suggests the inclusion into the WF algorithm of a parameter that scales the assumed input power spectra of the components, This parameter can be included in the all of the versions of the WF discussed above and is equivalent to assuming in Bayes' theorem a Gaussian prior of the form
In the use of this variant for the analysis of real data, is varied in order to obtain some desired signal-to-noise ratio in the reconstructed maps by artificially suppressing or enhancing the assumed power in the physical components as compared to the noise. Clearly, plays a similar role in the WF analysis to the parameter in the MEM. Thus, by making the appropriate changes to the calculation of the Bayesian value of in Appendix B, we may obtain an analogous expression to (26) that defines a Bayesian value for . Indeed, with the inclusion of the parameter , the WF method is simply the quadratic approximation to the MEM, as discussed in Section 2.5. However, even with the inclusion of the factor, we find that the corresponding reconstructions of non-Gaussian components are still somewhat poorer than for MEM.