

\subsection{\bf Error Calculation from $\chi^2$ 
Expansion}

\begin{slide}
\begin{itemize}
\item When we are at a minimum, $\Underline{F} = 0$ and Equation~(29) becomes
\[
\chi^2 (\Underline{$\alpha$}) = \chi^2 (\Underline{$\alpha$}_0) +
\Delta\Underline{$\alpha$}^T M\Delta\Underline{$\alpha$}
\]

\item or remembering that the likelihood
\begin{eqnarray}
L & = & \exp \left( - \frac{1}{2} \chi^2 \right) \nonumber \\
L & = & L (\alpha_0) \otimes \exp \left( - \frac{1}{2}
\Delta\Underline{$\alpha$}^T M\Delta\Underline{$\alpha$} \right)
\end{eqnarray}

\item Equation~(31) is expected Gaussian form for $L$.  

\item According to Section~3, $M$ is just inverse of error matrix, i.e., 
\[
\left\langle (\alpha^* - \tilde{\alpha})_j (\alpha^* - \tilde\alpha
)_k \right\rangle = M^{-1}_{jk}
\]

\begin{description}
\item{$\bullet$} where $\Underline{$\alpha$}^* = $ estimate from
minimizing $\chi^2$,
\item{$\bullet$} $\Underline{$\tilde\alpha$} = $ true parameters.
\end{description}

\item In particular, ``error'' $\sigma_j$ in $\alpha_j$ is
\begin{equation}
\sigma_j = \sqrt{(M^{-1})_{jj}}
\end{equation}
and the other elements in $M^{-1}$ are the much forgotten correlation
parameters.  

\item Note that $\Delta\alpha_j = \sigma_j$, all other
$\Delta\alpha_j = 0$, produces a $\chi^2$ change of 1.
\end{itemize}
\end{slide}

