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The general case with adaptive means for Gaussian prior factors
and hyperparameter energy yields an error functional
|
(432) |
Defining
|
(433) |
the stationarity equations of (432)
obtained from the functional derivatives with
respect to and hyperparameters
become
Inserting
Eq. (434) in Eq. (435) gives
|
(436) |
Eq.(436) becomes equivalent to the parametric
stationarity equation
(356) with vanishing prior term
in the deterministic limit of vanishing prior covariances ,
i.e., under the assumption
,
and for vanishing
.
Furthermore, a non-vanishing prior term in (356) can be
identified with the term .
This shows, that parametric methods can be considered
as deterministic limits of (prior mean) hyperparameter approaches.
In particular, a parametric solution can thus
serve as reference template ,
to be used within a specific prior factor.
Similarly,
such a parametric solution
is a natural initial guess for a nonparametric
when solving a stationarity equation by iteration.
If working with parameterized
extra prior terms Gaussian in some function can be included
as discussed in Section 4.2.
Then, analogously to templates for , also
parameter templates can be made adaptive
with hyperparameters .
Furthermore, prior terms
and
for the hyperparameters ,
can be added.
Including such additional error terms yields
and Eqs.(434) and (434) change to
where
,
,
,
denote derivatives with respect
to the parameters or , respectively.
Parameterizing and
the process
of introducing hyperparameters can be iterated.
Next: Unrestricted variation
Up: Adapting prior means
Previous: General considerations
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Joerg_Lemm
2001-01-21