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Gaussian priors for parameters
Up to now we assumed the prior to be given for a function
depending on and .
Instead of a prior in a function
also a prior in another not -dependent function
of the parameters can be given.
A Gaussian prior in
being a linear function of ,
results in a prior which is also Gaussian in
the parameters , giving a regularization term
|
(371) |
where =
is not an operator in a space of functions
but a matrix in the space of parameters .
The results of Section 4.1
apply to this case provided the following replacement is made
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(372) |
Similarly,
a nonlinear requires the replacement
|
(373) |
where
|
(374) |
Thus, in the general case where a Gaussian (specific) prior in
and is given,
or, including also non-zero template functions (means)
,
for and
as discussed in Section 3.5,
The and -terms of the energy
can be interpreted as corresponding to
a probability
,
(
),
or, for example,
to
with one of the two terms term
corresponding to a Gaussian likelihood
with -independent normalization.
The stationarity equation becomes
which defines ,
and for
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(379) |
for
.
Table 5:
Summary of stationarity equations.
For notations, conditions and comments see
Sections
3.1.1,
3.2.1,
3.3.2,
3.3.1,
4.1
and 4.2.
Variable |
Error |
Stationarity equation |
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Next: Linear trial spaces
Up: Parameterizing likelihoods: Variational methods
Previous: General parameterizations
  Contents
Joerg_Lemm
2001-01-21