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Referring to the discussion in Section 2.3
we show that Eq. (141) can
alternatively be obtained by ensuring normalization,
instead of using Lagrange multipliers,
explicitly by the parameterization
|
(146) |
and considering the functional
|
(147) |
The stationary equation for
obtained by setting the functional derivative
to zero yields again Eq. (141).
We check this, using
|
(148) |
and
|
(149) |
where
denotes a matrix,
and the superscript the transpose of a matrix.
We also note
that despite
|
(150) |
is not symmetric
because depends on and
does not commute with the non-diagonal
.
Hence, we obtain
the stationarity equation
of functional written in terms of
again Eq. (141)
|
(151) |
Here
is the -gradient of .
Referring to the discussion following Eq. (141)
we note, however, that solving for instead for
no unnormalized solutions
fulfilling are possible.
In case is in the zero space of
the functional corresponds to
a Gaussian prior in alone.
Alternatively, we may also directly
consider a Gaussian prior in
|
(152) |
with stationarity equation
|
(153) |
Notice, that expressing the density estimation problem in terms of ,
nonlocal normalization terms have not disappeared but are
part of the likelihood term.
As it is typical for density estimation problems,
the solution can be calculated in
-data space, i.e., in the space
defined by the of the training data.
This still allows to use a Gaussian prior structure
with respect to the -dependency
which is especially useful for classification problems
[236].
Next: The Hessians ,
Up: Gaussian prior factor for
Previous: Lagrange multipliers: Error functional
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Joerg_Lemm
2001-01-21