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\jairheading{4}{1996}{129-145}{11/95}{3/96}
\ShortHeadings{Active Learning with Statistical Models}
{Cohn, Ghahramani \& Jordan}
\firstpageno{129}


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\vspace{-0.2in}

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	\exp\left[-\frac{1}{2}({\bf x} - \mu_i)^T\Sigma_i^{-1}({\bf x} - \mu_i)
	\right]
	\label{joint-eqn}
\end{eqnarray}
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{\bf x} = \left[\begin{array}{c} x \\  y \end{array}\right] &
{\bf \mu}_i = \left[\begin{array}{c} \mu_{x,i} \\  \mu_{y,i} \end{array}\right]
&
{\bf \Sigma}_i = \left[\begin{array}{ll} \sigma^2_{x,i} & \sigma_{xy,i} \\ 
			 \sigma_{xy,i} & \sigma^2_{y,i} \end{array}\right].
\end{array}
\end{eqnarray*}
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{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{\hat{y},i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1432: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\hat{y}_i = \mu_{y,i} + \frac{\sigma_{xy,i}}{\sigma^2_{x,i}}(x - \mu_{x,i}),& &
\sigma^2_{\hat{y},i} = \frac{\sigma^2_{y|x,i}}{n_i}
	\left(1 + \frac{(x - \mu_{x,i})^2}{\sigma^2_{x,i}}\right).
	\label{mixture_est_eqn}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1436: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1438: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}n_i = \sum^m_{j=1} \frac{P(x_j, y_j | i)}{\sum_{k=1}^N P(x_j, y_j | k)}.
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1440: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_i \equiv h_i(x) = \frac{P(x | i)}{\sum_{j=1}^N P(x | j)},
\end{displaymath}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}P(x | i) = \frac{1}{\sqrt{2 \pi \sigma^2_{x,i}}}
	\exp\left[-\frac{(x - \mu_{x,i})^2}{2\sigma^2_{x,i}}\right].
	\label{marginal-eqn}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\hat{y}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1448: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\hat{y} = \sum_{i=1}^N h_i \: \hat{y}_i, & &
\sigma^2_{\hat{y}} = \sum_{i=1}^N \frac{h^2_i\sigma^2_{y|x,i}}{n_i}
	\left(1 + \frac{(x - \mu_{x,i})^2}{\sigma^2_{x,i}}\right),
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\hat{y}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1450: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{\hat{y}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1452: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{\hat{y}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1454: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\hat{y}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1458: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\hat{y}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1462: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1468: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1470: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1472: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu_{y,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1474: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{y,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1476: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma_{xy,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1478: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1480: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1482: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1484: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1486: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1488: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(\tilde{x},y|i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1492: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(x,y|i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1494: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}P(x,y|i) = P(\tilde{x},y|i)\frac{P(x|i)}{P(\tilde{x}|i)}. \label{correction-eqn}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(\tilde{x}|i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1496: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(x|i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1500: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1504: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1506: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left<\tilde{\sigma}^2_{\hat{y}}\right>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1508: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left<\tilde{\sigma}^2_{\hat{y}}\right>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1510: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{y}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1512: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1514: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{displaymath}P(\tilde{y}|\tilde{x}) = 
	\sum_{i=1}^N \tilde{h}_iP(\tilde{y}|\tilde{x}, i)
	= \sum_{i=1}^N \tilde{h}_i 
	{\cal N}(\hat{y}_i(\tilde{x}), \sigma^2_{y|x,i}(\tilde{x})),
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{h}_i \equiv h_i(\tilde{x})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1516: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{${\cal N}(\mu,
\sigma^2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1518: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1520: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1522: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$g_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1524: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(\tilde{y}|\tilde{x}, i)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1526: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{h}_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1528: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\left<\tilde{\sigma}^2_{\hat{y}}\right> & = & \sum_{i=1}^N 
	 \frac{h^2_i\left<\tilde{\sigma}^2_{y|x,i}\right>}{n_i + \tilde{h}_i}
   \left(1 + \frac{(x - \mu_{x,i})^2}{\sigma^2_{x,i}}\right)
	\label{mix-exp}
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\left<\tilde{\sigma}^2_{y,i}\right> =
	\frac{n_i\sigma^2_{y,i}}{n_i+\tilde{h}_i}
	+ \frac{n_i\tilde{h}_i  \left(\sigma^2_{y|\tilde{x},i} 
		+ (\hat{y}_i(\tilde{x}) - \mu_{y,i})^2\right)}
	     {(n_i+\tilde{h}_i)^2},
&&
\left<\tilde{\sigma}^2_{y|x,i}\right> =
   \left<\tilde{\sigma}^2_{y,i}\right> - 
	\frac{\left<\tilde{\sigma}^2_{xy,i}\right>}
	     {\sigma^2_{x,i}},
\\ 
\left<\tilde{\sigma}^{\mbox{ }}_{xy,i}\right> = 
	\frac{n_i\sigma_{xy,i}}
	     {n_i+\tilde{h}_i}
	+  \frac{n_i\tilde{h}_i  (\tilde{x} - \mu_{x,i})
			     (\hat{y}_i(\tilde{x}) - \mu_{y,i})}
			       {(n_i+\tilde{h}_i)^2},
&&
\left<\tilde{\sigma}^2_{xy,i}\right>  = 
   \left<\tilde{\sigma}_{xy,i}\right>^2
    + \frac{n_i^2\tilde{h}_i^2\sigma^2_{y|\tilde{x},i}
	(\tilde{x} - \mu_{x,i})^2}
	     {(n_i+\tilde{h}_i)^4}.
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1530: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1532: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\mu_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1534: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{x,i}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1536: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\tilde{\mu}_{x,i} = \frac{n_i\mu_{x,i} + \tilde{h}_i\tilde{x}}
                {n_i + \tilde{h}_i}, & &
\tilde{\sigma}^2_{x,i} = \frac{n_i\sigma^2_{x,i}}
                            {n_i+\tilde{h}_i}
                        + \frac{n_i\tilde{h}_i  (\tilde{x} - \mu_{x,i})^2}
                               {(n_i+\tilde{h}_i)^2}.
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1538: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1540: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\partial\left<\tilde{\sigma}^2_{\hat{y}}\right>/\partial
\tilde{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1542: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1544: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{figure}[bht]
\centerline{\psfig{figure=lwr_fig.ps,height=2in}}
\vspace{-0.2in}

\end{figure}
}

{\newpage
\clearpage
\samepage \begin{displaymath}h_i(x) \equiv h(x - x_i) = \exp(-k(x - x_i)^2),
\end{displaymath}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$h_i(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1558: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$n
= \sum_i h_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1560: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}& &\mu_x = \frac{\sum_i h_i x_i}{n}, \mbox{ } 
\sigma^2_{x} = \frac{\sum_i h_i  (x_i - \mu_x)^2}{n}, \mbox{ } 
\sigma_{xy} = \frac{\sum_i h_i  (x_i - \mu_x)(y_i - \mu_y)}{n}  \\ 
& &\mu_y = \frac{\sum_i h_i y_i}{n},  \mbox{ }
\sigma^2_{y} = \frac{\sum_i h_i  (y_i - \mu_y)^2}{n},  \mbox{ } 
\sigma^2_{y|x} = \sigma^2_{y} - \frac{\sigma^2_{xy}}{\sigma^2_{x}}.
\end{eqnarray*}
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\hat{y} =  \mu_y + \frac{\sigma_{xy}}{\sigma^2_x}(x - \mu_x),  & &
\sigma^2_{\hat{y}} = \frac{\sigma^2_{y|x}}{n^2}
	\left(\sum_i h_i^2 +
	\frac{(x - \mu_x)^2}{\sigma_x^2}
	\sum_i h_i^2\frac{(x_i - \mu_x)^2}{\sigma_x^2}\right)	
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \begin{figure}[t]
\centerline{\psfig{figure=smooth_fig.ps,height=2in}}
\vspace{-0.2in}

\end{figure}
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(x)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1578: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{\hat{y}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1582: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{\hat{y}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1588: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{y|x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1590: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{\hat{y}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1592: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{\hat{y}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1600: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

\stepcounter{subsection}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1602: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left<\tilde{\sigma}^2_{\hat{y}}\right>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1604: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$P(\tilde{y}|\tilde{x})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1606: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\hat{y}(\tilde{x})$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1608: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sigma^2_{y|\tilde{x}}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1610: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\left<\tilde{\sigma}^2_{\hat{y}}\right>$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1612: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{h}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1614: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\tilde{x}$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1616: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray}\left<\tilde{\sigma}^2_{\hat{y}}\right> =  
	\frac{\left<\tilde{\sigma}^2_{y|x}\right>}{(n + \tilde{h})^2}
	\left[\sum_i h_i^2 + \tilde{h}^2 + 
	\frac{(x - \tilde{\mu}_x)^2}{\tilde{\sigma}_x^2}
	\left(\sum_i h_i^2\frac{(x_i - \tilde{\mu}_x)^2}
		{\tilde{\sigma}_x^2}
	 + \tilde{h}^2\frac{(\tilde{x} - \tilde{\mu}_x)^2}
		{\tilde{\sigma}_x^2}\right)\right].\label{lwr-exp} & &
\end{eqnarray}
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sum_i h_i^2 (x_i - \mu_x)^2 = \sum_i h_i^2 x^2_i +
\mu^2_x\sum_i h_i^2 - 2 \mu_x\sum_i h_i^2 x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1618: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sum_i h_i^2 x^2_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1620: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$\sum_i h_i^2 x_i$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1622: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
\samepage \begin{eqnarray*}\left<\tilde{\sigma}^2_{y|x}\right> =
   \left<\tilde{\sigma}^2_{y}\right> - 
	\frac{\left<\tilde{\sigma}^2_{xy}\right>}
	     {\tilde{\sigma}^2_{x}},
& &
\left<\tilde{\sigma}^2_{y}\right> =
	\frac{n\sigma^2_{y}}
	     {n+\tilde{h}}
	+ \frac{n\tilde{h}  \left(\sigma^2_{y|\tilde{x}} + (\hat{y}(\tilde{x}) - \mu_{y})^2\right)}
	     {(n+\tilde{h})^2},
\\ 
\tilde{\mu}_{x} = \frac{n\mu_{x} + \tilde{h}\tilde{x}}
		{n + \tilde{h}},
& &
\left<\tilde{\sigma}_{xy}\right> = 
	\frac{n\sigma_{xy}}
	     {n+\tilde{h}}
	+  \frac{n\tilde{h}  (\tilde{x} - \mu_{x})
			     (\hat{y}(\tilde{x}) - \mu_{y})}
			       {(n+\tilde{h})^2},
\\ 
\tilde{\sigma}^2_{x} = \frac{n\sigma^2_{x}}
			    {n+\tilde{h}}
			+ \frac{n\tilde{h}  (\tilde{x} - \mu_{x})^2}
			       {(n+\tilde{h})^2},
& &
\left<\tilde{\sigma}^2_{xy}\right>  = 
   \left<\tilde{\sigma}_{xy}\right>^2
    + \frac{n^2\tilde{h}^2\sigma^2_{y|\tilde{x}}(\tilde{x} - \mu_{x})^2}
	     {(n+\tilde{h})^4}.
\end{eqnarray*}
}

\stepcounter{section}
{\newpage
\clearpage
\samepage \setbox\sizebox=\hbox{$(\Theta_1, \Theta_2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1624: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
}

{\newpage
\clearpage
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$(\theta_1, \theta_2)$}\lthtmltypeout{latex2htmlSize :tex2html_wrap_inline1630: \the\ht\sizebox::\the\dp\sizebox.}\box\sizebox
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{\newpage
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\samepage \begin{figure}[hbt]
\centerline{\psfig{figure=arm_fig.ps,height=2in}}
\vspace{-0.2in}

\end{figure}
}

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\stepcounter{subsection}
{\newpage
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\samepage \begin{figure}[hbt]
\centerline{
	\parbox{3.0in}{\psfig{figure=em_noise_var.ps,width=3in}}
	\parbox{3.0in}{\psfig{figure=em_noise.ps,width=3in}}}
\vspace{-0.25in}

\end{figure}
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{\newpage
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\centerline{
	\parbox{3in}{\psfig{figure=new_lwr_noise_var.ps,width=3in}}
	\parbox{3in}{\psfig{figure=new_lwr_noise.ps,width=3in}}}
\vspace{-0.25in}

\end{figure}
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{\newpage
\clearpage
\samepage \begin{table}\begin{center}
\begin{tabular}{|l|l|l|l|} \hline
	& Training & Evaluating Reference & Evaluating Candidates \\  \hline
Mixture	& $3.9 + 0.05m$ sec & 15000 $\mu$sec      & 1300 $\mu$sec \\  
LOESS	& -                 & $92+9.7m$ $\mu$sec   &  $58+0.16m$ $\mu$sec \\ 
\hline
\end{tabular}
\end{center}

\end{table}
}

\stepcounter{section}
\appendix
\stepcounter{section}
{\newpage
\clearpage
\samepage \begin{tabular}{ll}
& General \\  \hline
$X$		& input space \\ 
$Y$		& output space \\ 
$x$		& an arbitrary point in the input space \\ 
$y$		& true output value corresponding to input $x$ \\ 
$\hat{y}$	& predicted output value corresponding to input $x$ \\ 
$x_i$		& ``input'' part of example $i$ \\ 
$y_i$		& ``output'' part of example $i$ \\ 
$m$		& the number of examples in the training set \\ 
$\tilde{x}$	& specified input of a query \\ 
$\tilde{y}$	& the (possibly not yet known) output of query $\tilde{x}$ \\ 
$\sigma^2_{\hat{y}}$ & estimated variance of $\hat{y}$ \\  
$\tilde{\sigma}^2_{\hat{y}}$ & new variance of $\hat{y}$, after
example $(\tilde{x}, \tilde{y})$ has been added \\  
$\left<\tilde{\sigma}^2_{\hat{y}}\right>$ & the expected value of
	 $\tilde{\sigma}^2_{\hat{y}}$ \\ 
$P(x)$		& the (known) natural distribution over $x$ \\  
& \\ 
& Neural Network \\  \hline
$w$		& a weight vector for a neural network \\ 
$\hat{w}$	& estimated ``best'' $w$ given a training set \\ 
$f_{\hat{w}}()$	& function computed by neural network given $\hat{w}$ \\ 
$S^2$		& average estimated noise in data, used as an estimate
		for $\sigma^2_y$  \\ 
& \\ 
& Mixture of Gaussians \\  \hline
$N$		& total number of Gaussians \\ 
$g_i$		& Gaussian number $i$ \\ 
$n_i$		& total point weighting attributed to Gaussian $i$ \\ 
$\mu_{x,i}$	& estimated $x$ mean of Gaussian $i$ \\ 
$\mu_{y,i}$	& estimated $y$ mean of Gaussian $i$ \\ 
$\sigma^2_{x,i}$& estimated $x$ variance of Gaussian $i$ \\ 
$\sigma^2_{y,i}$& estimated $y$ variance of Gaussian $i$ \\ 
$\sigma_{xy,i}$ & estimated $xy$ covariance of Gaussian $i$ \\ 
$\sigma^2_{y|x,i}$ & estimated $y$ variance of Gaussian $i$, given $x$ \\ 
$P(x, y|i)$	& joint distribution of input-output pair given Gaussian $i$ \\ 
$P(x|i)$	& distribution $x$ being given Gaussian $i$ \\ 
$h_i$		& weight of a given point that is attributed to Gaussian $i$ \\ 
$\tilde{h}_i$	& weight of new point $(\tilde{x}, \tilde{y})$ that is
		attributed to Gaussian $i$ \\ 
& \\ 
& Locally Weighted Regression \\  \hline
$k$		& kernel smoothing parameter \\ 
$h_i$		& weight given to example $i$ by kernel centered at $x$ \\ 
$n$		& sum of weights given to all points by kernel \\ 
$\mu_x$		& mean of inputs, weighted by kernel centered at $x$ \\ 
$\mu_y$		& mean of outputs, weighted by kernel centered at $x$ \\ 
$\tilde{h}$	& weight of new point $(\tilde{x}, \tilde{y})$ given 
		kernel centered at $x$ \\ 
\end{tabular}
}

\stepcounter{section}

\end{document}
