

\subsection{\bf The Real World and $\chi^2$
Minimization}
\subsection{\bf Example of Two Variables---III}

\begin{slide}
\begin{itemize}
\item Unfortunately, there is no reason to believe that this large
shift is well estimated as assumptions that lead to expansion break
down.  

\begin{description}
\item{$\bullet$} Terms like $B_2 (\Delta\alpha_1)^2 \Delta\alpha_2$ are
quite likely to be big.
\end{description}

\item We see that estimate in Equation~(30) is false and one will
typically jump out of valley and find
\[
\chi^2 \left( \Underline{$\alpha$}_0 + \Delta\Underline{$\alpha$} \right)
>> \chi^2 \left( \Underline{$\alpha$}_0 \right)
\]

\item The method will diverge!

\item In the above example, the estimate for $\Delta\alpha_1$ was
valid and not spoiled by $\Delta\alpha_2$ estimate.  

\item In a real example, these will be mixing between $\alpha_1$ and
$\alpha_2$ due to nonzero $\frac{\partial^2 \chi}{\partial\alpha_1
\partial\alpha_2}$.

\item Both shifts $\Delta\alpha_1$, $\Delta\alpha_2$ will be large and
specious.
\end{itemize}
\end{slide}

