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The optimal regression function under squared-error loss
-- for Gaussian regression identical to the log-loss
of density estimation --
is the predictive mean.
For mixture model (12)
one finds, say for fixed ,
|
(27) |
with
mixture coefficients
The component means
and
the likelihood of can be calculated analytically
[17,14]
and
|
(30) |
where
and
is the projection
of the covariance
into the
-dimensional space
for which training data are available.
( is the number of data with distinct
-values.)
The stationarity equation
for a maximum a posteriori approximation
with respect to
is at this stage found from
(28,30)
|
(33) |
where
=
+
.
Notice that Eq.(33)
differs from Eq.(22)
and requires only to deal with
the
-matrix
.
The coefficient
=
for set to its maximum posterior value
is of form
(23)
with the replacements
,
.
Next: High and low temperature
Up: Prior mixtures
Previous: Maximum a posteriori approximation
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Joerg_Lemm
1999-12-21