Next: Example: Square root of
Up: General Gaussian prior factors
Previous: General Gaussian prior factors
  Contents
The general case
In the previous sections we studied
priors consisting of a factor (the specific prior) which was Gaussian
with respect to or
and additional normalization (and non-negativity) conditions.
In this section we consider the situation where the
probability
is expressed in terms of a function .
That means, we assume
a, possibly non-linear, operator
= which maps the function
to a probability.
We can then formulate a learning problem in terms
of the function , meaning that
now represents the hidden variables
or unknown state of Nature .3Consider the case of a specific prior
which is Gaussian in ,
i.e., which has a specific prior energy quadratic in
|
(186) |
This means we are lead to error functionals of the form
|
(187) |
where we have skipped the -independent part of the -terms.
In general cases also
the non-negativity constraint has to be implemented.
To express
the functional derivative of functional (187)
with respect to we define
besides the diagonal matrix =
the Jacobian, i.e., the matrix of derivatives
=
with matrix elements
|
(188) |
The matrix
is diagonal
for point-wise transformations, i.e., for
=
.
In such cases we use to denote the vector of diagonal elements
of
.
An example is the previously discussed transformation
for which
= .
The stationarity equation for functional (187) becomes
|
(189) |
and for existing
=
(for nonexisting inverse
see Section 4.1),
|
(190) |
From Eq. (190)
the Lagrange multiplier function can be found by integration,
using the normalization condition
= ,
in the form
=
for
.
Thus, multiplying Eq. (190) by
yields
|
(191) |
is now eliminated by
inserting Eq. (191) into Eq. (190)
|
(192) |
A simple iteration procedure,
provided
exists,
is suggested by writing
Eq. (189) in the form
|
(193) |
with
|
(194) |
Table 2
lists constraints to be implemented explicitly
for some choices of .
Table 2:
Constraints for specific choices of
|
|
constraints |
|
|
norm |
non-negativity |
|
|
-- |
non-negativity |
|
|
norm |
-- |
|
|
-- |
-- |
|
|
boundary |
monotony |
|
Next: Example: Square root of
Up: General Gaussian prior factors
Previous: General Gaussian prior factors
  Contents
Joerg_Lemm
2001-01-21