
\subsection{\bf IMPORTANCE Sampling}
\subsection{\bf Basic Theory}

\begin{slide}
\begin{itemize}
\item (iii) Of course, it is obvious from (NI.16) that Monte Carlo integration
gets better as standard deviation $\sigma$ gets smaller.
\item Using definition of $\sigma^2$ (or remembering its name
``variance''), it follows that method is best when $f(x)$ is slowly
varying over domain.
\item For instance, if $f(x) \equiv$ constant, $\sigma =0$ and method
is exact.
\item Importance sampling (``sampling where integrand important'') is a
technique for reducing variation of $f(x)$.
\item Thus if
\begin{eqnarray*}
I & = & \int f(x)\,dx \\
\mbox{change variables}\hspace{2.5em} y & = & q(x)\,, \hspace{0.5em} 
dy = q'(x)\,dx \\
\mbox{put}\hspace{1em} {f(x)\over{q'(x)}} & = & F(y) \\
\mbox{then}\hspace{2.5em} I & = & \int F(y)\,dy. 
\end{eqnarray*}
\item We now apply Monte Carlo method generating $y$ uniformly.
\end{itemize}
\end{slide}

