

\subsection{\bf Warning Example of Misuse}
\subsection{\bf of Likelihood Method}

\begin{slide}
\begin{itemize}
\item Consider probability distribution appropriate for polarization
experiment:
\[
p (\varphi , \alpha ) = 
\frac{(1 + \alpha\cos\varphi)}{2\pi}
\hspace{1em} : -1 \le \alpha \le 1.
\]

\item If we had but one event (a very small experiment!)
\[
L(\alpha ) = \frac{(1 + \alpha\cos\varphi_1)}{2\pi}
\]
\begin{tabular}{lll}
$\bullet$ if \hspace{1em}& $\cos\varphi_1 > 0$ \hspace{1em} & : $\alpha=1$
(i.e., end of range) \\
&& maximizes $L$ \\
$\bullet$ or \hspace{1em}& $\cos\varphi_1 < 0$ \hspace{1em} & : $\alpha=-1$
(other end of range) \\
&& maximizes $L$
\end{tabular}

\item If you have two events:
\[
L(\alpha) = \frac{(1 + \alpha (\cos \varphi_1 + \cos \varphi_2 ) +
\alpha^2 \cos \varphi_1 \cos \varphi_2 )}{2\pi}
\]

\item This has a maximum at end of range, again, unless $\cos\varphi_1
\cos\varphi_2 < 0$.
\end{itemize}
\end{slide}

