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We are going to approximate the distribution in
equation 3 by replacing the discrete labels,
, with
continuous weights vectors,
:
|
(9) |
where
} and
is the continuous weights
vector at voxel . Equation 9 only
approximates equation 3 if we apply certain
constraints to the continuous weights vectors. If we choose a
prior on the continuous weights vector, , with the
constraints that
and
, then as
tends to delta functions at and
, then equation 9 will tend
to equation 3. Therefore, to apply these
constraints the prior we use is:
where:
and
, where crucially
is a deterministic
relationship by which and
are
related by the logistic transform:
|
(12) |
The normalising constant in the logistic transform
ensures that the condition
is met. This expression also ensures
that
, if and only if
. Figure 1 shows how the
logistic transform produces an approximation to the delta
functions as gets smaller. We fix the value of
to 0.05 whilst bounding
, this ensures
that we get the desired approximation to delta functions at 0 and
1, whilst ensuring that we can compute
without causing overflow.
To summarise, we now have two vectors of continuous weights at each voxel,
and
.
are weights which have a prior on them which
is uniform on the real line.
We then use the logistic transform to deterministically map
the weights
to at each voxel.
Then, are the continuous weights which
represent approximations to
the discrete labels
with delta functions at 0 and 1.
Figure 1:
Consider that we have the number of classes as .
[top] shows samples from the
prior of
,
(samples from
are
similar). [bottom] shows the samples from
( is similar), which the samples from the prior
of
and
transform to under the logistic transform with
different values of
(equation 12). Hence, it can be seen
how this produces a prior for
, which approximates the
desired delta functions at 0 and as gets
smaller.
|
Next: Non-spatial with Class Proportions
Up: Continuous Weights Mixture Model
Previous: Continuous Weights Mixture Model