

\subsection{\bf Derivative Calculation for}
\subsection{\bf Maximum Likelihood---I}

\begin{slide}
\begin{itemize}
\item Let's do the same thing for maximum likelihood method.
\[
-\ln L = - \sum^N_{i=1} \ln q_i
\]

\item where $q_i$ are probability of our $i=1\ldots N$ events.  

\item Assume $q_i$
are normalized ${\displaystyle\int\limits} q(\alpha ) dx = 1$ as in
Equation~(25).  

\item This normalization is crucial but even when it does
not seem to be true as in our subtle modifications in Section~4.3 with
the ``generalized'' maximum likelihood method, one can easily modify
the analysis, and show it is still valid.

\item We get in normalized $q_i$ case,
\[
\frac{-\partial^2 \ln L}{\partial\alpha^2} = S_1 + S_2 
\]
\item with
\begin{eqnarray*}
S_1 & = & \sum^N_{i=1} \frac{1}{q_i} \frac{\partial^2
q_i}{\partial\alpha^2} \\
&& \\
S_2 & = & \sum^N_{i=1} \frac{1}{q_i^2} \left( \frac{\partial
q_i}{\partial\alpha} \right)^2
\end{eqnarray*}
\end{itemize}
\end{slide}

