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Example: Square root of
We already discussed the cases
with
,
and with ,
.
The choice
yields the common -normalization condition over
|
(195) |
which is quadratic in ,
and ,
,
.
For real
the non-negativity condition is automatically
satisfied [82,211].
For =
and a negative Laplacian inverse covariance = ,
one can relate the corresponding Gaussian prior
to the Fisher information
[38,211,207].
Consider, for example, a problem with fixed
(so can be skipped from the notation and one can write )
and a -dimensional .
Then one has, assuming the necessary differentiability conditions
and vanishing boundary terms,
|
(196) |
|
(197) |
where
is the Fisher information,
defined as
|
(198) |
for the family with location parameter
vector .
A connection to quantum mechanics can be found
considering the case without training data
|
(199) |
having the homogeneous stationarity equation
|
(200) |
For -independent
this is an eigenvalue equation.
Examples include the quantum mechanical
Schrödinger equation
where corresponds to the system Hamiltonian
and
|
(201) |
to its ground state energy.
In quantum mechanics Eq. (201) is the basis
for variational methods (see Section 4)
to obtain approximate solutions for ground state energies
[55,197,27].
Similarly, one can take
for bounded from above by
with the normalization
|
(202) |
and
,
,
= .
Next: Example: Distribution functions
Up: General Gaussian prior factors
Previous: The general case
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Joerg_Lemm
2001-01-21