

\subsection{\bf General Maximum Likelihood}
\subsection{\bf Methods---I}

\begin{slide}
\begin{itemize}
\item The above is glib in many ways---especially in its treatment of
errors.  

\begin{description}
\item{$\bullet$} We will repair this later with real live theorems.  
\item{$\bullet$} First, we state the obvious generalization.
\end{description}

\item The general maximum likelihood method is to put $P(\alpha\vert T)
= 1$ in Equation~(14)

\item and form $L(\alpha ) = \Pi^n_{i=1} p\left( x_i , \alpha \right)$

\item for any measurement $x_i$ whose probability $p\left( x_i, \alpha
\right)$ depends on value $x_i$ and theoretical parameter(s) $\alpha$.

\item The maximum likelihood estimate of $\alpha$ is the value
$\alpha^*$ where $L(\alpha^* )$ is a maximum.

\item The error in estimate is the standard deviation of the distribution 
of $L$ as a function of $\alpha$, i.e.,
\[
L(\alpha ) = \exp \left\lbrace \mbox{const.} -
\frac{1}{2\Delta\alpha^2} \left( \alpha - \alpha^* \right)^2
\right\rbrace
\]
\end{itemize}
\end{slide}

