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It are the limits of large and small
which make the introduction of this additional parameter useful.
The reason being that the high temperature limit
gives the convex case,
and statistical mechanics provides us
with high and low temperature expansions.
Hence, we study the high temperature and low temperature limits
of Eq. (551).
In the high temperature limit
the exponential factors become -independent
|
(553) |
In case one chooses
=
one has to replace
by .
The high temperature solution becomes
|
(554) |
with (generalized) `complete template average'
|
(555) |
Notice that corresponds to the minimum of the quadratic
functional
|
(556) |
Thus, in the infinite temperature limit
a combination of quadratic priors by OR is
effectively replaced by a combination by AND.
In the low temperature limit
we have,
assuming
=
,
|
(557) |
|
(558) |
meaning that
|
(559) |
Henceforth,
all (generalized) `component averages' become solutions
|
(560) |
with
|
(561) |
provided the
fulfill the stability condition
|
(562) |
i.e.,
|
(563) |
where
|
(564) |
That means
single components become solutions at zero temperature
in case their (generalized) `template variance' ,
measuring the discrepancy
between data and prior term, is not too large.
Eq. (551)
for can also be expressed by the
(potential) low temperature solutions
|
(565) |
Summarizing, in the high temperature limit
the stationarity equation (548)
becomes linear with a single solution being essentially
a (generalized) average of all template functions.
In the low temperature limit the single component solutions become stable
provided their (generalized) variance corresponding to their minimal error
is small enough.
Next: Equal covariances
Up: Prior mixtures for regression
Previous: Prior mixtures for regression
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Joerg_Lemm
2001-01-21