Maximum Entropy analysis

6.1 Maximum Entropy analysis

With such a large increase in the amount of data available from CMB experiments it is becoming increasingly important to improve the analysis techniques used. With multifrequency observations of the same patch of sky to high precision (the satellite will cover the full sky) it should be possible to extract information on the foregrounds and leave a ‘clean’ map of the CMB. One technique that has been shown to be robust in the analysis of a number of different experiments is Maximum Entropy ([38

], [44 ] and [55 ]).

If we start with Bayes’ theorem which states, given a hypothesis $H$ and some data $D$ the posterior probability $Pr(H |D )$ is the product of the likelihood $Pr (D |H )$ and the prior probability $Pr (H )$ , normalised by the evidence $Pr(D )$ ,

If the instrumental noise on each frequency channel is Gaussian-distributed, then the probability distribution of the noise is a multivariate Gaussian. Assuming the expectation value of the noise to be zero at each observing frequency, the likelihood is therefore given by

where the $2 χ$ misfit statistic has been introduced.

We now have to decide the form of the prior probability. Let us consider a discretised image $hj$ consisting of $L$ cells, so that $j = 1,...,L$ ; we may consider the $hj$ as the components of an image vector $H$ . If we base the derivation of the prior on purely information theoretic considerations (subset independence, coordinate invariance and system independence) we are naturally led to the Maximum Entropy Method (MEM). It may be shown [85 ] the prior probability takes the form

where the dimensional constant $α$ depends on the scaling of the problem and may be considered as a regularising parameter, and $M$ is a model vector to which $H$ defaults in the absence of any data.

In standard applications of the maximum entropy method, the image $H$ is taken to be a positive additive distribution (PAD). Nevertheless, the MEM approach can be extended to images that take both positive and negative values by considering them to be the difference of two PADS, so that

where $U$ and $V$ are the positive and negative parts of $H$ respectively. In this case, the cross entropy is given by

where $ψj = [h2j + 4mujmvj ]1∕2$ and $Mu$ and $Mv$ are separate models for each PAD. The global maximum of the cross entropy occurs at $H = Mu − Mv$ . The most probable image $H$ is then just the result from finding the maximum probability or, equivalently, the minimum of $χ2 − αS$ . This can be done by using any known minimising routine.

It can be shown [38 ] that the Lagrange multiplier $α$ is completely defined in a Bayesian way and any prior correlation information can also be incorporated into the analysis. Also, the assignment of errors is straightforward in the Fourier domain where all the pixels in the discretised image will be independent.

Hobson et al. [38 ] simulated data taken by the Planck Surveyor satellite and used MEM to reconstruct the underlying CMB and foregrounds. They used six input maps (the CMB, thermal and kinetic SZ, dust emission, free-free emission and synchrotron emission) to make up the data and then added Gaussian noise to each frequency. After using MEM with the Bayesian value for $α$ and giving the algorithm the average power spectra of each channel, it was found that features in all six maps were recovered. Without any prior power spectrum information it was found that only the kinetic SZ was not recovered and all others were recovered to some degree (the CMB and dust were almost indistinguishable from the input maps with residual errors of $6 μK$ and $2 μK$ per pixel respectively). Figure 20 shows the results from MEM as compared to the input maps for the case with assumed average power spectrum. It is easily seen that MEM reconstructs both the Gaussian CMB and the non-Gaussian thermal SZ effect very well.

Figure 20: The left hand side shows the input maps used in the Planck simulations for a CDM simulation of the CMB and the thermal SZ effect. The right hand side shows the reconstructions obtained by MEM. It is easily seen that MEM does a very good job at reconstructed these two components. For comparison, the grey scales on the input maps are the same as on the reconstructed maps.