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Unrestricted variation
To get a first understanding of the approach (432)
let us consider the extreme example of
completely unrestricted -variations.
In that case the template function itself
represents the hyperparameter.
(Such function hyperparameters or hyperfields
are also discussed in Sect. 5.6.)
Then,
and
Eq. (435) gives
=
(which for invertible
is solved uniquely by ),
resulting according to Eq. (229) in
|
(441) |
The case of a completely free prior mean
is therefore equivalent to a situation without prior.
Indeed, for invertible
,
projection of Eq. (436) into the -data space
by of Eq. (260)
yields
|
(442) |
where
=
is invertible
and
.
Thus for for which are available
|
(443) |
is concentrated on the data points.
Comparing this with solutions of Eq. (228),
for = fixed,
we see that adaptive means tend to lower the influence
of prior terms.
Next: Regression
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Joerg_Lemm
2001-01-21