\batchmode
\documentstyle[jair,twoside,11pt,theapa]{article}
\makeatletter
\input epsf 
\jairheading{10}{1999}{1-15}{10/98}{5/99}
\ShortHeadings{Variational Probabilistic Inference and QMR-DT}
{Jaakkola \& Jordan}
\firstpageno{1}


\renewcommand {\[}{\begin{eqnarray}}

\renewcommand {\]}{\end{eqnarray}}
\input{psfig.sty}


\makeatother
\newenvironment{tex2html_wrap}{}{}
\newwrite\lthtmlwrite
\def\lthtmltypeout#1{{\let\protect\string\immediate\write\lthtmlwrite{#1}}}%
\newbox\sizebox
\begin{document}
\pagestyle{empty}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mathop {\mathgroup \symoperators log}\nolimits P(f^+ | \xi )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline845: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^*(\xi )$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline853: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
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{\newpage
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}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{\psfig{figure=figures/qmr-toy.ps,height=1.95in}}

\label{f:qmr}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f,d) &=& P(f|d)P(d) = \left[ \prod_i P(f_i|d)\right] 
\left[ \prod_j P(d_j) \right]
\label{gjoint}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f_i|d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline875: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f_i=0|d) &=& P(f_i=0|L)\prod_{j\in \pi_i} P(f_i=0|d_j)
\label{negf}
\\ 
&=& (1-q_{i0}) \prod_{j\in \pi_i} (1-q_{ij})^{d_j} \\ 
&\equiv& e^{\:-\theta_{i0}-\sum_{j\in \pi_i} \theta_{ij} d_j},
\label{negfe}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\pi_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline877: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$f_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline879: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_{ij} = P(f_i=1|d_j=1)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline881: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_{i0} = P(f_i=1|L)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline885: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta_{ij} = -\log(1-q_{ij})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline887: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
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\samepage \setbox\sizebox=\hbox{$f_i=1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline895: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f_{i'}=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline897: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^{+}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline899: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^{-}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline903: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$f_i=0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline905: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$P(d_j|f^+,f^-)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline907: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^+$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline909: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^-$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline911: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$d_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline915: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(d|f^+,f^-)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline917: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$P(d|f^-)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline919: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^-|d)P(d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline921: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^-|d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline923: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(d_j) \leftarrow P(d_j|f^-)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline931: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(d_j|f^+)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline933: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+|d)P(d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline935: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(d_j|f^+) \propto \sum_{d\setminus d_j} P(f^+|d)P(d)
\label{marg}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$d\setminus d_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline939: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+|d)P(d) &=& \left[ \prod_i P(f^+_i|d)\right] \left[ \prod_j P(d_j) \right]
\\ 
&=& 
\left[ \prod_i \left( 1 - e^{\:-\theta_{i0}-\sum_{j} \theta_{ij} d_j} \right)
\right] \left[
\prod_j P(d_j)
\right]
\label{joint}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+_i|d) = 1-P(f^-|d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline943: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{displaymath}f(x) = \min_\xi\{\: \xi^T x - f^*(\xi)\: \}
\label{dualmin}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}f^*(\xi) = \min_x\{\: \xi^T x - f(x)\: \}.
\label{conjmin}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^*$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline963: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline967: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}f(x) \leq \xi^T x - f^*(\xi).
\label{ubound}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(a + \sum_j z_j)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline971: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \setbox\sizebox=\hbox{$z_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline975: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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\samepage \setbox\sizebox=\hbox{$i\in\{1,2,\ldots,n\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline977: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(a + \sum_j q_j x_j) \geq \sum_j q_j f(a + x_j)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline983: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
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}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$0 \leq q_j \leq 1$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline989: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$a + \sum_j q_j x_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline993: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}f(\, a+\sum_j z_j \,) &=& f(\, a+\sum_j q_j \frac{z_j}{q_j} \,)  \\ 
&\geq& \sum_j q_j\,f(\,a+ \frac{z_j}{q_j} \,)
\label{e:genericlower}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline995: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_j = z_j/\sum_k z_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1001: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+_i|d) &=& 1 - e^{\:-\theta_{i0}-\sum_{j} \theta_{ij} d_j}
= e^{\:\log( 1 - e^{-x})}
\label{e:noisy-or}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x=\theta_{i0}+\sum_{j} \theta_{ij} d_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1015: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(x) = \log( 1 - e^{-x})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1017: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}f(x) \leq \xi x - f^*(\xi)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}f^*(\xi) = -\xi\log\xi+(\xi+1)\log(\xi+1)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+_i|d) &=& e^{\:f(\theta_{i0}+\sum_{j}\theta_{ij} d_j)}
\\ 
&\leq& e^{\: \xi_i(\theta_{i0}+\sum_{j} \theta_{ij} d_j) - f^*(\xi_i)}
\label{var-qmr}
\\ 
&\equiv& P(f^+_i|d,\xi_i).
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+_i|d,\xi_i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1029: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+_i|d,\xi_i) &=& e^{\: \xi_i(\theta_{i0}+\sum_{j} \theta_{ij} d_j) - f^*(\xi_i)} \\ 
&=& e^{\:\xi_i\theta_{i0}-f^*(\xi_i)} 
\prod_j \left[ e^{\: \xi_i\theta_{ij}} \right]^{d_j}.
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+_i|d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1037: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f(\theta_{i0}+\sum_{j}\theta_{ij} d_j)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1039: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta_{ij} d_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1045: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\theta_{i0}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1049: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+_i|d) &=& e^{\,f(\,\theta_{i0}+\sum_{j}\theta_{ij} d_j\,)} 
\\ 
&\geq& e^{\,\sum_{j} q_{j|i} 
f\left(\,\theta_{io} + \frac{\theta_{ij} d_j}{q_{j|i}}\,\right)
}
\\ 
&=& e^{\, \sum_{j} q_{j|i} \left[\, d_j\, f\left(\,\theta_{io} +
\frac{\theta_{ij}}{q_{j|i}}\,\right) + (1-d_j)\,
f\left(\,\theta_{io}\,\right)\, \right]
}
\\ 
&=& e^{\,
\sum_{j} q_{j|i}\, d_j\, \left[\, f\left(\,\theta_{io} +
\frac{\theta_{ij}}{q_{j|i}}\,\right) - f\left(\,\theta_{io}\,\right)\, \right]
+ f\left(\,\theta_{io}\,\right)
}
\label{e:varlower}
\\ 
&\equiv& P(f^+_i|d,q_{\cdot|i})
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_{\cdot|i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1051: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+_i|d,q_{\cdot|i}) \leq P(f^+_i|d) \leq P(f^+_i|d,\xi_i).
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1075: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+|\xi)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1077: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+) = \sum_d P(f^+|d) P(d) \leq
\sum_d P(f^+|d,\xi)P(d) \equiv P(f^+|\xi)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\delta_i = P(f^+|\xi) - P(f^+|\xi\setminus\xi_i)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+|\xi\setminus\xi_i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1083: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$i^{th}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1085: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\delta_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1087: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{\psfig{figure=figures/b-br-1.eps,height=1.75in}}

\label{f:order}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+) = \sum_{d} P(f^+|d)P(d) \leq \sum_{d} P(f^+|d,\xi)P(d) \equiv P(f^+|\xi)
\label{e:upper}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+) = \sum_{d} P(f^+|d)P(d) \geq \sum_{d} P(f^+|d,q)P(d) \equiv P(f^+|q)
\label{e:lower}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+,d_j|\xi)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1115: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$j^{th}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1119: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+,d_j) = \sum_{d\setminus d_j}P(f^+|d)P(d)
\leq \sum_{d\setminus d_j}P(f^+|d,\xi)P(d) \equiv P(f^+,d_j|\xi)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+|d,\xi)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1123: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+,d_j|q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1125: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+,d_j) = \sum_{d\setminus d_j}P(f^+|d)P(d)
\geq \sum_{d\setminus d_j}P(f^+|d,q)P(d) \equiv P(f^+,d_j|q)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{P(f^+,d_j|q)}{P(f^+,\bar{d}_j|\xi)+P(f^+,d_j|q)}
\leq P(d_j|f^+) \leq 
\frac{P(f^+,d_j|\xi)}{P(f^+,d_j|\xi)+P(f^+,\bar{d}_j|q)},
\label{eq:interval-bounds}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\bar{d}_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1127: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{table}[ht]
\begin{center}
\begin{tabular}{|c|c|c|} \hline
case & \# of pos. findings & \# of neg. findings \\  \hline
1  & 20 & 14 \\  \hline
2  & 10 & 21 \\  \hline
3  & 19 & 19 \\  \hline
4  & 19 & 33 \\  \hline 
\end{tabular}
\vskip6pt

\label{t:cases} 
\end{center}
\end{table}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\log P(f^+ | \xi)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1133: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
(a) \psfig{figure=figures/b-8.eps,height=1.75in} \hspace{12pt}
(b) \psfig{figure=figures/b-12.eps,height=1.75in} }

\label{f:likelihood-bounds}
\end{figure}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+|q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1137: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
(a) \psfig{figure=figures/8-corr.eps,height=1.75in} \hspace{12pt}
(b) \psfig{figure=figures/12-corr.eps,height=1.75in} }

\label{f:scatter-of-var-posteriors}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
\psfig{figure=figures/corr.eps,height=1.95in} \hspace{12pt}}

\label{f:var-vs-shwe-cooper}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
(a) \psfig{figure=figures/8-lo.eps,height=1.75in} \hspace{12pt}
(b) \psfig{figure=figures/8-lo-fn.eps,height=1.75in} }

\label{f:all-8}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
(a) \psfig{figure=figures/12-lo.eps,height=1.75in} \hspace{12pt}
(b) \psfig{figure=figures/12-lo-fn.eps,height=1.75in} }

\label{f:all-12}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
\psfig{figure=figures/12-lo.eps,height=1.75in} \hspace{12pt}}

\label{f:rankplot-shwe-cooper}
\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
(a) \psfig{figure=figures/ul-s.eps,height=1.75in} \hspace{12pt}
(b) \psfig{figure=figures/s1s2.eps,height=1.75in} }

\label{f:cpc-all}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
\psfig{figure=figures/s1s2-select.eps,height=1.75in} }

\label{f:cpc-select}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{\psfig{figure=figures/all-12-s8.eps,height=1.75in}}

\label{f:cpc-tpfp-select}
\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{
(a) \psfig{figure=figures/ul-16-exact.eps,width=1.7in} \hspace{12pt}
(b) \psfig{figure=figures/ul-16-select.eps,width=1.7in} \hspace{12pt}
(c) \psfig{figure=figures/ul-16-all.eps,width=1.7in} \hspace{12pt}}

\label{f:gap-hist}
\end{figure}
}

\stepcounter{section}
\appendix
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$X = R^n$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1213: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\{ (x,y): y\geq f(x) \}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1225: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{figure}[ht]
\centerline{\psfig{figure=figures/dual.eps,height=1.95in}}

\label{f:dual}
\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\mbox{all}\; (x,y) \;\mbox{such that} \; x^T\xi - y - \mu \leq 0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1241: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(x,y)\in epi(f)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1245: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x^T\xi - y - \mu \leq 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1247: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x^T\xi - f(x) -
\mu \leq 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1249: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$y\geq f(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1251: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$x \in X$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1253: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\max_{x\in X} \{\: x^T\xi - f(x) - \mu \:\}\leq 0,
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\mu &\geq& \max_{x\in X}\{\: x^T\xi - f(x)\: \}
\label{dualfrom-f}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^*(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1263: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}x^T\xi \leq f(x) + f^*(\xi)
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}f(x) = \max_{\xi\in \Xi}\{\: x^T\xi - f^*(\xi)\: \} 
\label{f-fromdual}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\max$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1269: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\min$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1271: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+) = \sum_d P(f^+|d)P(d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1287: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+|d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1289: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+) \leq \sum_d P(f^+|d,\xi)P(d) \equiv P(f^+|\xi).
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+|\xi) &=& \sum_d \left[ \prod_{i\leq m} P(f_i^+|d,\xi_i) \right]
\left[ \prod_{i>m} P(f_i^+|d) \right] \prod_j P(d_j)
\\ 
&\propto& E\left\{ \prod_{i\leq m} P(f_i^+|d,\xi_i) \right\},
\label{opt:vl}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+|\xi) &\propto& E\left\{ \prod_{i\leq m} 
e^{\: \xi_i(\theta_{i0}+\sum_{j} \theta_{ij} d_j) - f^*(\xi_i)}
\right\}
\\ 
&=& e^{\:\sum_{i\leq m} \left(\xi_i\theta_{i0}-f^*(\xi_i)\right)\:}
E\left\{ e^{\: \sum_{j,\,i\leq m}\xi_i \theta_{ij} d_j} \right\}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\log P(f^+|\xi) = C + \sum_{i\leq m} \left(\xi_i\theta_{i0}-f^*(\xi_i)\right)
+ \log E\left\{ e^{\: \sum_{j,i\leq m}\xi_i \theta_{ij} d_j} \right\}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$f^*(\xi_i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1303: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$-f^*(\xi_i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1307: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\log E\left\{ e^{\: \sum_{j,i\leq m}\xi_i \theta_{ij} d_j} \right\}
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\nabla \log P(f^+|\xi) = 0$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1311: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1313: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{\partial}{\partial \xi_k}\log P(f^+|\xi) &=& \theta_{k0}+\log \frac{\xi_k}{1+\xi_k}
+ E\left\{ \sum_j \theta_{kj} d_j \right\}
\\ 
\frac{\partial^2}{\partial^2 \xi_k}\log P(f^+|\xi) &=&
\frac{1}{\xi_k} - \frac{1}{1+\xi_k}
+ {\mbox Var}\left\{ \sum_j \theta_{kj} d_j \right\},
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(d|f^+,\xi)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1315: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+|d,q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1325: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(f^+) \geq \sum_d P(f^+|d,q)P(d) \equiv P(f^+|q).
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\log P(f^+|q)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1343: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}E\left\{\, \log P(f^+|d,q)P(d)\, \right\} = 
\sum_i 
E\left\{\, \log P(f^+_i|d,q_{\cdot|i}) \,\right\} + \mbox{constant},
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q^{old}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1345: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(d|f^+,q^{old}) \propto P(f^+|d,q^{old})P(d)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1349: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+_i|d,q_{\cdot|i})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1351: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}E\left\{\, \log P(f^+_i|d,q_{\cdot|i}) \,\right\} &=& 
\sum_{j} q_{j|i}\, E\{d_j\}\, \left[\, f\left(\,\theta_{io} +
\frac{\theta_{ij}}{q_{j|i}}\,\right) - f\left(\,\theta_{io}\,\right)\, \right]
\nonumber\\ 
&& + f\left(\,\theta_{io}\,\right) 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_{j|i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1353: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$E\{ d_j \}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1355: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}q_{j|i} &\leftarrow& E\{d_j\} \left[\, q_{j|i}\, 
f\left(\,\theta_{io}+\frac{\theta_{ij}}{q_{j|i}}\,\right) 
-\theta_{ij}\, f'\left(\,\theta_{io}+\frac{\theta_{ij}}{q_{j|i}}\,\right) 
- q_{j|i}\, f(\,\theta_{io}\,)
\,\right] \nonumber\\ 
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(f^+|d,q^{old})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1361: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\xi_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1375: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$X = \sum_{j,i\leq m}\xi_i \theta_{ij}
d_j$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1377: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\log E\left\{ e^{\,X} \right\} = \log \sum_i p_i\, e^{\,X_i} = f(\vec{X})
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\vec{X} = \{X_1 \ldots X_n\}^T$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1381: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$X_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1385: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$p_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1387: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$X_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1391: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}\frac{\partial}{\partial X_k} f(\vec{X}) = \frac{ p_k e^{\,X_k} }{\sum_i p_i\,
e^{\,X_i}}  \equiv q_k
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_k$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1393: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\cal H}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1395: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}{\cal H}_{kl} = \frac{\partial^2}{\partial X_k \partial X_l} f(\vec{X}) = \delta_{kl} q_k - q_k q_l
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{displaymath}\vec{Z}^T {\cal H} \vec{Z} = 
\sum_k q_k Z^2_k - (\sum_k q_k Z_k)(\sum_l q_l Z_l)
= {\mbox Var}\{Z\} \geq 0
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\mbox Var}\{Z\}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1399: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$Z_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1403: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$q_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1405: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}

\end{document}
