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Prior mixtures for regression
For regression it is especially useful to introduce
an inverse temperature multiplying the terms
depending on , i.e., likelihood and prior.4As in regression is represented by the regression function
the temperature-dependent error functional becomes
|
(544) |
with
|
(545) |
|
(546) |
some hyperprior energy
,
and
with some constant .
If we also maximize with respect to
we have to include the (-independent)
training data variance
where
=
is the variance of the training data at .
In case every appears only once vanishes.
Notice that includes a contribution from the data points
arising from the -dependent normalization
of the likelihood term.
Writing the stationarity equation
for the hyperparameter separately,
the corresponding three stationarity conditions
are found as
As is only a one-dimensional parameter
and its density can be quite non-Gaussian
it is probably most times more informative
to solve for varying values of
instead to restrict to a single
`optimal' .
Eq. (548)
can also be written
|
(551) |
with
being thus still a nonlinear equation for .
Subsections
Next: High and low temperature
Up: Non-Gaussian prior factors
Previous: Prior mixtures for density
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Joerg_Lemm
2001-01-21