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To statistically model the measurement process, a number of issues
must be addressed. First, there is the physics describing the
measurement process. This, however, is
assumed to be set down in Eq. (1). Second, a number of details
related to the measurement process itself, must be properly
incorporated. These include the manner in which the data was
collected, e.g., as a rectangular grid of counts as is the case for
astronomical imaging with a solid state detector, as well as the
characteristics of the noise, etc. Finally, a number of decisions must be
made regarding how Eq. (1) is to be inverted mathematically.
Common assumptions here are that the image can be represented by a grid
of numbers and that the integral represented in Eq. (1) can be
approximated by a discrete sum over this grid. Each of these aspects
of the problem, i.e., the physics, the particulars of the measurement,
and the mathematical assumptions are part of the model, , which links
the image and the data.
The Bayesian approach to inverting Eq. (1) is to use Bayes'
Theorem to develop a formula for the most probable value of .
This begins by factoring the joint probability
distribution of the data, image, and model triplet,
,
and
,
i.e.,
:
where is the probability of
given that
is known. [Bayes' Theorem states that
.] Equating the
various terms in Eq. (2), we find
From Eq. (4) we can find the M.A.P. (Maximum A Posteriori) image using
where is the model from the M.A.P. image/model pair. (The
proportionality in Eq. (5) should really be an
``approximately-proportional'' symbol since we have dropped
some terms [e.g.
] and we have
assumed that the peak in the probability distribution
is representative of a typical sample from the posterior probability
distribution. While this is usually the case, this is not absolutely
guaranteed.)
Eq. (3) is the typical starting point of Bayesian image
reconstruction in which one wishes to determine the M.A.P.
image, i.e. the image which maximizes . (The
M.A.P. image, of course, is only one of several choices for the ``best
image''. Another sensible choice might be the average image,
.)
Eqs. (4) and (5) are our preferred prescription for Bayesian image reconstruction in which both the image and the model are varied simultaneously to obtain the best combined image/model solution, e.g., the M.A.P. image/model pair. Eq. (4) might be used directly to find the optimal image, i.e., the image paired with the model in the M.A.P. image/model pair. Alternatively, the M.A.P. image can be found using Eq. (5).
The significance of the terms in the above equations are well known.
The first term, , is a goodness-of-fit (GOF) quantity. A
typical choice for
is to use
,
where
is the chi-square of the residuals. This form for
the GOF statistic allows the best unbiased estimate for the image to be
determined.
However, it produces a value for
which is uncharacteristically
small (i.e.,
). Another approach that we have explored is to
use
where
is the actual chi-squared distribution.
This choice for the GOF criterion recognizes that the most probable
value for
is not zero, but depends on
the number of random variables. This GOF criterion also seems to produce
quite good image reconstructions and to avoid
problems associated with ``over-fitting'' the data.
The terms and
are ``priors''. Since they do not
depend on the data they can be decided a priori. The first of
these, i.e.,
, is normally termed the image prior and
expresses the a priori probability of an image given the
model. The second, i.e.,
, we have termed the image/model prior, and expresses the a priori probability of
both
and
. In GOF image reconstruction the prior is ignored
or is effectively set equal to unity, i.e., there is no prior bias
concerning the image or the model. In Maximum Entropy (ME) image
reconstruction, the image prior is based upon ``phase space volume'' or
counting arguments and the prior is expressed as
, where
is the entropy of the image and
is an adjustable
constant that is used to weight the relative importance of the GOF and
image prior.
The final quantity, , is termed the ``Evidence'' for the model.
Actually, we normally refer to
as the Evidence, but
is
proportional to this quantity under the common assumption that
and
(there generally is not
an a priori
manner for choosing between valid data sets nor between ``sensible''
models.) Choosing models on the basis of the Evidence allows one to
``peak-up'' on more favorable solutions.