If we start with Bayes' theorem which states, given a
hypothesis
H
and some data
D
the posterior probability
is the product of the likelihood
and the prior probability
, normalised by the evidence
,
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
misfit statistic has been introduced.
We now have to decide the form of the prior probability. Let
us consider a discretised image
consisting of
L
cells, so that
; we may consider the
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 [92] the prior probability takes the form
where the dimensional constant
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
and
and
are separate models for each PAD. The global maximum of the
cross entropy occurs at
. The most probable image
H
is then just the result from finding the maximum probability or,
equivalently, the minimum of
. This can be done by using any known minimising routine.
It can be shown [57] that the Lagrange multiplier
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.
[57] 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
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
K and
K 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.
![]() |
The Cosmic Microwave Background
Aled W. Jones and Anthony N. Lasenby http://www.livingreviews.org/lrr-1998-11 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |