

\subsection{\bf Maximum Likelihood Principle}
\subsection{\bf Illustrated by Measurement Errors---II}

\begin{slide}
\begin{itemize}
\item The likelihood function for whole experiment of $n$ events is 
defined as:
\begin{eqnarray*}
L\left( x_1 \ldots x_n \right) & = & \prod^n_{i=1} c \exp \left\lbrace - 
\frac{1}{2\sigma^2_0} \left( x_i - \alpha_0 \right)^2 \right\rbrace \\
& = & \tilde{c} \exp \left\lbrace - \frac{1}{2\sigma^2_0} \sum_i
\left( x_i - \alpha_0 \right)^2 \right\rbrace
\end{eqnarray*}

\item $c$ and $\tilde{c}$ are calculable constants to normalize
probability.  This is---up to a constant---the probability of the $x_i$
given,

\begin{description}
\item{$\bullet$} the true length $\alpha_0$,
\item{$\bullet$} and certain theoretical assumptions or
prejudices that we represent as $T$.
\end{description}

\[
L \left( x_i \ldots x_n \right) = Pr \left\lbrace x_1 \ldots x_n \vert
\alpha_0 , T \right\rbrace
\]
using conditional probability notation.  

\item $T$ is, for instance, the notion that distribution is Gaussian
and the value of $\sigma_0$ taken (the latter being calculable from the
strength of earthquake {\em or\/} amount of beer drunk).
\end{itemize}
\end{slide}

