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Kernels for $L$

For $E_L$ we have the truncated equation

\begin{displaymath}
L
= {{\bf C}} N.
\end{displaymath} (688)

Normalizing the exponential of the solution gives
\begin{displaymath}
L(x,y) = \sum_i^n {{\bf C}} (x,y;x_i,y_i)
- \ln \int \! dy^\prime \, e^{\sum_i^n {{\bf C}} (x,y^\prime ;x_i,y_i)} ,
\end{displaymath} (689)

or
\begin{displaymath}
L = {\bf C} N -\ln {\bf I}_X {\bf e}^{{\bf C}N}
.
\end{displaymath} (690)

Notice that normalizing $L$ according to Eq. (689) after each iteration the truncated equation (688) is equivalent to a one-step iteration with uniform $P^{(0)}$ = $e^{L^{(0)}}$ according to
\begin{displaymath}
L^1 = {{\bf C}} N + {\bf C}{\bf P}^{(0)} \Lambda_X,
\end{displaymath} (691)

where only $({\bf I}-{\bf C}{\bf K})L$ is missing from the nontruncated equation (683), because the additional $y$-independent term ${\bf C}{\bf P}^{(0)} \Lambda_X$ becomes inessential if $L$ is normalized afterwards.

Lets us consider as example the choice ${{\bf C}}$ = $-{\bf H}^{-1}(\phi^{(0)})$ for uniform initial $L^{(0)}=c$ corresponding to a normalized $P$ and ${{\bf K}}L^{(0)}$ = $0$ (e.g., a differential operator). Uniform $L^{(0)}$ means uniform $P^{(0)}=1/v_y$, assuming that $v_y = \int \!dy$ exists and, according to Eq. (137), $\Lambda_X$ = $N_X$ for ${{\bf K}}L^{(0)}$ = $0$. Thus, the Hessian (161) at $L^{(0)}$ is found as

\begin{displaymath}
{\bf H}(L^{(0)})
=
- \left( {\bf I} - \frac{{\bf I}_X}{v_y}...
...I}_X}{v_y} \right)
\frac{{\bf N}_X}{v_y}
= -{{\bf C}}^{-1}
,
\end{displaymath} (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

\begin{displaymath}
L^\epsilon_{\rm emp} = \ln P^\epsilon_{\rm emp}
,
\end{displaymath} (693)

with $P^\epsilon_{\rm emp}$ 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,
\begin{displaymath}
L^{(0)} =
{\bf C} L^\epsilon_{\rm emp}
- \ln {\bf I}_X {\bf e}^{{\bf C} L^\epsilon_{\rm emp}}
.
\end{displaymath} (694)

Similarly, one may let a kernel ${\bf C}$, or its normalized version $\tilde {\bf C}$ defined below in Eq. (698), act on $P_{\rm emp}$ first and then take the logarithm
\begin{displaymath}
L^{(0)} =
\ln (\tilde {\bf C} P^\epsilon_{\rm emp})
.
\end{displaymath} (695)

Because already $\tilde {\bf C} P_{\rm emp}$ is typically nonzero it is most times not necessary to work here with $P^\epsilon_{\rm emp}$. Like in the next section $P_{\rm emp}$ may be also be replaced by $\tilde P_{\rm emp}$ as defined in Eq. (238).


next up previous contents
Next: Kernels for Up: Initial configurations and kernel Previous: Truncated equations   Contents
Joerg_Lemm 2001-01-21