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For we have the truncated equation
|
(688) |
Normalizing the exponential of the solution gives
|
(689) |
or
|
(690) |
Notice that normalizing
according to Eq. (689)
after each iteration
the truncated equation (688)
is equivalent to a one-step iteration with uniform =
according to
|
(691) |
where only
is missing from the
nontruncated equation (683),
because the additional -independent term
becomes inessential if is normalized afterwards.
Lets us consider as example the choice
=
for uniform initial
corresponding to a normalized
and
=
(e.g., a differential operator).
Uniform means uniform ,
assuming that
exists
and, according to Eq. (137),
= for
= .
Thus, the Hessian (161) at
is found as
|
(692) |
which can be invertible due to the presence of the second term.
Another possibility is to start with an
approximate empirical log-density, defined as
|
(693) |
with
given in Eq. (239).
Analogously to Eq. (686),
the empirical log-density may for example also be smoothed
and correctly normalized again,
resulting in an initial guess,
|
(694) |
Similarly, one may let a kernel ,
or its normalized version
defined below in Eq. (698),
act on first and then take the logarithm
|
(695) |
Because already
is typically
nonzero it is most times not necessary to work here
with
.
Like in the next section
may be also be replaced by
as defined in Eq. (238).
Next: Kernels for
Up: Initial configurations and kernel
Previous: Truncated equations
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Joerg_Lemm
2001-01-21