Next: Covariances and invariances
Up: General Gaussian prior factors
Previous: Example: Square root of
  Contents
Instead in terms of the probability density function,
one can formulate the prior in terms of its integral,
the distribution function. The density is then recovered
from the distribution function by differentiation,
|
(203) |
resulting in a non-diagonal
.
The inverse of the derivative operator is
the integration operator =
with matrix elements
|
(204) |
i.e.,
|
(205) |
Thus, (203)
corresponds to the transformation of (-conditioned)
density functions in (-conditioned)
distribution functions
= , i.e.,
=
.
Because
is positive (semi-)definite
if is,
a specific prior which is Gaussian in the distribution function
is also Gaussian in the density .
becomes
|
(206) |
Here the derivative of the -function is defined
by formal partial integration
|
(207) |
Fixing
the variational derivative
is not needed.
The normalization condition for
becomes
for the distribution function =
the boundary condition
,
.
The non-negativity condition for corresponds
to the monotonicity condition
,
,
and to
,
.
Next: Covariances and invariances
Up: General Gaussian prior factors
Previous: Example: Square root of
  Contents
Joerg_Lemm
2001-01-21