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\jairheading{13}{2000}{155-188}{6/00}{10/00}
  \ShortHeadings{Adaptive Importance Sampling in Bayesian Networks}
  {Cheng \& Druzdzel}
  \firstpageno{155}

\hyphenation{Ba-ye-sian}

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$ \approx$%
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\renewedcommand{baselinestretch}{1.0}
\begin{minipage}[t]{5.3in}
\begin{enumerate}
\item
  Order the nodes according to their topological order.
\item
  Initialize importance function ${\mbox{\rm Pr}}^0({\bf X}\backslash{\bf E})$  using some heuristic methods, initialize weight $w^0$, set the
  desired number of samples $m$\  and the updating interval $l$, initialize the score arrays for every node.
\item
  $k\leftarrow 0$, $T\leftarrow \emptyset$, $w_{TScore}\leftarrow
  0$, $w_{\mbox{\it sum}}\leftarrow 0$\item
  {\bf for} $i\leftarrow 1$\  {\bf to} $m$\  {\bf do}
\item
  \quad {\bf if} ($i\;\mbox{mod}\;l==0$) {\bf then}
\item
  \qquad $k\leftarrow k+1$\item
  \qquad Update the importance function ${\mbox{\rm Pr}}^k({\bf X}\backslash{\bf E})$  and $w^k$\  based on $T$.
  \par
\quad {\bf end if}
\item
  \quad ${\bf s}_i\leftarrow$\  generate a sample according to
  ${\mbox{\rm Pr}}^k({\bf X}\backslash{\bf E})$\item
  \quad $T\leftarrow T\cup \{{\bf s}_i\}$\item
  \quad $w_{\mbox{\it iScore}}\leftarrow$    Score (${\bf s}_i$, ${\mbox{\rm Pr}}({\bf X}\backslash{\bf E},{\bf e})$,
    ${\mbox{\rm Pr}}^k({\bf X}\backslash{\bf E})$, $w^k$)
\item
  \quad $w_{TScore}\leftarrow w_{TScore}+w_{iScore}$  \par
\quad{\footnotesize (Optional: add $w_{iScore}$\  to the corresponding entry of every score array)}
\item
  \quad $w_{\mbox{\it sum}}\leftarrow w_{\mbox{\it sum}}+w^k$  \par
{\bf end for}
\item
  Output estimate of ${\mbox{\rm Pr}}({\bf E})$\  as $w_{TScore}/w_{\mbox{\it sum}}$  \par
{\footnotesize (Optional: Normalize the score arrays for every node)}
\end{enumerate}
\end{minipage}}}%
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$\displaystyle {\frac{{\mbox{\rm Pr}}({\bf s}_i,{\bf e})}{{\mbox{\rm Pr}}^k({\bf s}_i)}}$%
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$ \propto$%
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$ \widehat{\sigma}^{k}_{}$%
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$ \rho$%
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$ \rho^{*}_{}$%
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$\displaystyle \rho$%
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$\displaystyle {\frac{{\mbox{\rm Pr}}({\bf X}\backslash{\bf E},{\bf E}={\bf e})}{{\mbox{\rm Pr}}({\bf E}={\bf e})}}$%
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$ \bf E)$%
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$\scriptstyle \prime$%
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$\displaystyle \notin$%
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$\displaystyle \Rightarrow$%
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$ \notin$%
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$ \Box$%
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\fbox{
\parbox{5.3in}{
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\begin{minipage}[t]{5.3in}
{\em Input:} Initialized importance function ${\mbox{\rm Pr}}^0({\bf
X}\backslash{\bf E})$, learning rate $\eta (k)$.
\\
{\em Output:} An estimated importance function ${\mbox{\rm Pr}}^S({\bf
X}\backslash{\bf E})$.
\par
{\bf for} stage $k\leftarrow 0$\  {\bf to} $S$\  {\bf do}
\begin{enumerate}
\item
  Sample $l$\  points ${\bf s}_1^k$, ${\bf s}_2^k$, \ldots,
  ${\bf s}_l^k$\  independently according to the current importance
  function ${\mbox{\rm Pr}}^k({\bf X}\backslash{\bf E})$.
\item
  For every node $X_i$\  such that $X_i\in{\bf X}\backslash{\bf E}$  and $X_i\notin{\mbox{\rm Anc}}({\bf E})$\  count score sums corresponding
  to $\{x_i,{\mbox{\rm pa}}(X_i),{\bf e}\}$\  and estimate
  ${\mbox{\rm Pr}}^{\prime}(x_i|{\mbox{\rm pa}}(X_i),{\bf e)}$\  based on
  ${\bf s}_1^k$, ${\bf s}_2^k$, \ldots, ${\bf s}_l^k$.
\item
  Update ${\mbox{\rm Pr}}^k({\bf X}\backslash{\bf E})$\  according to
  the following formula:
  \begin{eqnarray*}
  \lefteqn{{\mbox{\rm Pr}}^{k+1}(x_i|{\mbox{\rm pa}}(X_i),{\bf e}) =} \\
    && {\mbox{\rm Pr}}^k(x_i|{\mbox{\rm pa}}(X_i),{\bf e})+\eta(k)
    \cdot\left({\mbox{\rm Pr}}^{\prime}(x_i|{\mbox{\rm pa}}(X_i),
    {\bf e})-{\mbox{\rm Pr}}^k(x_i|{\mbox{\rm pa}}(X_i),{\bf e})\right)
  \end{eqnarray*}
\end{enumerate}
{\bf end for}
\end{minipage}}}%
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$\displaystyle \eta$%
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$\displaystyle \left(\vphantom{\frac{b}{a}}\right.$%
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$ \mu_{Z}^{}$%
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$ \sigma_{Z}^{2}$%
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$ \overline{Z}$%
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$\displaystyle {\frac{\sigma_{Z}/\sqrt{n}}{\mu_z}}$%
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$\displaystyle \Phi_{Z}^{-1}$%
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$\displaystyle {\frac{\delta }{2}}$%
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$ \mu_{z}^{}$%
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$ \bf a_i$%
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$ \sum_{i=1}^{k}$%
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$\displaystyle \sqrt{\frac{1}{\sum_{X_{i}\in {\bf N}\backslash {\bf E}}n_{i}} \sum_{X_{i}\in {\bf N}\backslash{\bf E}}\sum_{j=1}^{n_{i}}({\mbox{\rm Pr}}^{\prime
}(x_{ij})-{\mbox{\rm Pr}}(x_{ij}))^{2}}$%
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$ \bf N$%
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$ \leq$%
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$ \geq$%
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$\displaystyle {\frac{{\mbox{\rm Pr}}(x_i|{{\mbox{\rm pa}}}(X_i))
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     (x_i|{{\mbox{\rm pa}}}(X_i),{\bf e})}{1+k}}$%
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{\newpage\clearpage
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$ \widehat{{\mbox{\rm Pr}}}_{\mbox{\it current}}^{}$%
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$ \bf e$%
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$ \rho^{\prime}_{}$%
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$ \sigma_{{\mbox{\rm Pr}}({\bf e})}^{2}$%
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$ \sim$%
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\bgroup \egroup
\bgroup \egroup
\bgroup \egroup
\bgroup \egroup
\bgroup \egroup
\bgroup \egroup
\bgroup \egroup
\bgroup \egroup
\bgroup \egroup
\bgroup \egroup

\end{document}
