www.pudn.com > Gaussian.zip > images.pl, change:2004-06-07,size:78613b


# LaTeX2HTML 2002-2-1 (1.70)
# Associate images original text with physical files.


$key = q/ensuremathmathbf{x}circlearrowleft{calN}(mathbf{0},mathbf{I});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.gif"
 ALT="$ \ensuremath\mathbf{x}\circlearrowleft {\cal N}(\mathbf{0},\mathbf{I})$">|; 

$key = q/ensuremathmathbf{x}inensuremathmathbb{R}^d;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.gif"
 ALT="$ \ensuremath\mathbf{x}\in \ensuremath\mathbb{R}^d$">|; 

$key = q/displaystylep(x|ensuremathboldsymbol{Theta}_1)>p(x|ensuremathboldsymbol{Theta}_2nsuremathboldsymbol{Theta}_1)>logp(x|ensuremathboldsymbol{Theta}_2),;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="329" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.gif"
 ALT="$\displaystyle p(x\vert\ensuremath\boldsymbol{\Theta}_1) > p(x\vert\ensuremath\b...
...emath\boldsymbol{\Theta}_1) > \log p(x\vert\ensuremath\boldsymbol{\Theta}_2),
$">|; 

$key = q/displaystylefrac{P(q_k^{(i)}|ensuremathboldsymbol{Theta}^{(i)})cdotp(ensuremathm^{(i)})}{p(ensuremathmathbf{x}_n|ensuremathboldsymbol{Theta}^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.gif"
 ALT="$\displaystyle \frac{P(q_k^{(i)}\vert\ensuremath\boldsymbol{\Theta}^{(i)})
\cdot...
...}^{(i)})}
{p(\ensuremath\mathbf{x}_n\vert\ensuremath\boldsymbol{\Theta}^{(i)})}$">|; 

$key = q/n=1,cdots,N;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.gif"
 ALT="$ n=1,\cdots,N$">|; 

$key = q/displaystyleg_{(mu,Sigma)}(x)=frac{1}{sqrt{2pi}^dsqrt{detleft(Sigmaright)}},e^{-frac{1}{2}(x-mu)^{mathsfT}Sigma^{-1}(x-mu)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img7.gif"
 ALT="$\displaystyle g_{(\mu,\Sigma)}(x) = \frac{1}{\sqrt{2\pi}^d
\sqrt{\det\left(\Sigma\right)}}   e^{-\frac{1}{2} (x-\mu)^{\mathsf T}
\Sigma^{-1} (x-\mu)}
$">|; 

$key = q/p(ensuremathmathbf{x}_i|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.gif"
 ALT="$ p(\ensuremath\mathbf{x}_i\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/Sigma_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.gif"
 ALT="$ \Sigma_2$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_2=left[{array}{cc}8000&00&18500{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_2 = \left[ \begin{array}{cc}
8000 & 0 \\\\
0 & 18500
\end{array} \right]
$">|; 

$key = q/Theta;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img86.gif"
 ALT="$ \Theta$">|; 

$key = q/f_{text{slashoslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.gif"
 ALT="$ f_{\text{/o/}}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/{calN}_1:;Theta_1=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&00&8000{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.gif"
 ALT="$ {\cal N}_1: \; \Theta_1 = \left(
\left[\begin{array}{c}730  1090\end{array}\right],
\left[\begin{array}{cc}8000 & 0  0 & 8000\end{array}\right]
\right)$">|; 

$key = q/displaystyled_k(x_n);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.gif"
 ALT="$\displaystyle d_k(x_n)$">|; 

$key = q/p(X|q_k,Theta)P(q_k|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.gif"
 ALT="$ p(X\vert q_k,\Theta)
P(q_k\vert\Theta)$">|; 

$key = q/displaystyleP(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.gif"
 ALT="$\displaystyle P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$">|; 

$key = q/displaystylelogP(q_k|X,ensuremathboldsymbol{Theta})proptologp(X|q_k,ensuremathboldsymbol{Theta})+logP(q_k|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.gif"
 ALT="$\displaystyle \log P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \propto \log p(...
...ensuremath\boldsymbol{\Theta}) + \log P(q_k\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/p(X,Q|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.gif"
 ALT="$ p(X,Q\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/P(q_k|X,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.gif"
 ALT="$ P(q_k\vert X,\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyle{calL}(Theta)=logp(X|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.gif"
 ALT="$\displaystyle {\cal L}(\Theta) = \log p(X\vert\Theta)$">|; 

$key = q/P(q_k^{(i)}|x_n,Theta^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.gif"
 ALT="$ P(q_k^{(i)}\vert x_n,\Theta^{(i)})$">|; 

$key = q/displaystyleP(q_k|x)geqP(q_l|x),foralllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.gif"
 ALT="$\displaystyle P(q_k\vert x) \geq P(q_l\vert x), \forall l \neq k$">|; 

$key = q/{calN}_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.gif"
 ALT="$ {\cal N}_2$">|; 

$key = q/[530,1000]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.gif"
 ALT="$ [530,1000]^{\mathsf T}$">|; 

$key = q/ensuremathboldsymbol{Sigma}_{text{slasheslash}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.gif"
 ALT="$ \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}$">|; 

$key = q/X_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.gif"
 ALT="$ X_1$">|; 

$key = q/|mathbf{A}-mathbf{B}|_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.gif"
 ALT="$ \Vert\mathbf{A}-\mathbf{B}\Vert _2$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_{k}^{(i+1)}=frac{sum_{n=1}^{N}P(q_k^{(i)(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N}...
...P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $">|; 

$key = q/p(X|q_k,Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.gif"
 ALT="$ p(X\vert q_k,\Theta)$">|; 

$key = q/{ensuremathboldsymbol{mu}_{text{slashaslash}},ensuremathboldsymbol{mu}_{text{slau}_{text{slashoslash}},ensuremathboldsymbol{mu}_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.gif"
 ALT="$ \{\ensuremath\boldsymbol{\mu}_{\text{/a/}},
\ensuremath\boldsymbol{\mu}_{\tex...
...emath\boldsymbol{\mu}_{\text{/o/}},
\ensuremath\boldsymbol{\mu}_{\text{/y/}}\}$">|; 

$key = q/y=sqrt{Sigma},x+mu;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.gif"
 ALT="$ y = \sqrt{\Sigma}   x + \mu$">|; 

$key = q/displaystylequadP(q_k|X,ensuremathboldsymbol{Theta})geqP(q_j|X,ensuremathboldsymbol{Theta}),quadforalljneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img96.gif"
 ALT="$\displaystyle \quad P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \geq P(q_j\vert X,\ensuremath\boldsymbol{\Theta}),
\quad\forall j \neq k
$">|; 

$key = q/displaystyleforallk,;;P(q_k|X,Theta)proptop(X|q_k,Theta)P(q_k|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="233" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.gif"
 ALT="$\displaystyle \forall k, \;\; P(q_k\vert X,\Theta) \propto p(X\vert q_k,\Theta) P(q_k\vert\Theta)
$">|; 

$key = q/q_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.gif"
 ALT="$ q_k$">|; 

$key = q/displaystyle=sum_{k=1}^{K}sum_{n=1}^{N}P(q_k|ensuremathmathbf{x}_n,ensuremathbolymbol{Theta})logp(ensuremathmathbf{x}_n|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.gif"
 ALT="$\displaystyle = \sum_{k=1}^{K} \sum_{n=1}^{N} P(q_k\vert\ensuremath\mathbf{x}_n...
...bol{\Theta}) \log p(\ensuremath\mathbf{x}_n\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/ensuremathmathbf{x}_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.gif"
 ALT="$ \ensuremath\mathbf{x}_i$">|; 

$key = q/displaystyleP(q_k|X,ensuremathboldsymbol{Theta})=frac{p(X|q_k,ensuremathboldsymb(q_k|ensuremathboldsymbol{Theta})}{p(X|ensuremathboldsymbol{Theta})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="216" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.gif"
 ALT="$\displaystyle P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) = \frac{p(X\vert q_k...
...k\vert\ensuremath\boldsymbol{\Theta})}{p(X\vert\ensuremath\boldsymbol{\Theta})}$">|; 

$key = q/[400,1800]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.gif"
 ALT="$ [400,1800]^{\mathsf T}$">|; 

$key = q/displaystyle{calL}(ensuremathboldsymbol{Theta})=logp(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="122" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.gif"
 ALT="$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) = \log p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/p(X|ensuremathboldsymbol{Theta})=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.gif"
 ALT="$ p(X\vert\ensuremath\boldsymbol{\Theta})=$">|; 

$key = q/displaystyleensuremathboldsymbol{mu}=left[{array}{c}7301090{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\mu}= \left[ \begin{array}{c} 730  1090 \end{array} \right]
$">|; 

$key = q/displaystyle=logsum_{k=1}^{K}P(q_k|X,ensuremathboldsymbol{Theta})p(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.gif"
 ALT="$\displaystyle = \log \sum_{k=1}^{K} P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyleensuremathboldsymbol{mu}_{k}^{(i+1)}=frac{sum_{n=1}^{N}ensuremathmat(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} \e...
...P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $">|; 

$key = q/q_k^{(i)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.gif"
 ALT="$ q_k^{(i)}$">|; 

$key = q/mu_i=E(x_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img11.gif"
 ALT="$ \mu_i = E(x_i)$">|; 

$key = q/logleft(detleft(ensuremathboldsymbol{Sigma}right)right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.gif"
 ALT="$ \log \left( \det\left(\ensuremath\boldsymbol{\Sigma}\right)
\right)$">|; 

$key = q/ensuremathmathbf{x};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img20.gif"
 ALT="$ \ensuremath\mathbf{x}$">|; 

$key = q/Theta_kinTheta;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.gif"
 ALT="$ \Theta_k \in \Theta$">|; 

$key = q/logp(X|q_k,Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.gif"
 ALT="$ \log p(X\vert q_k,\Theta)$">|; 

$key = q/{calN}_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.gif"
 ALT="$ {\cal
N}_1$">|; 

$key = q/displaystyleSigma=left({array}{*{4}{c}}c_{11}&c_{12}&cdots&c_{1n}c_{21}&c_{22}&cc_{2n}vdots&vdots&ddots&vdotsc_{n1}&c_{n2}&cdots&c_{nn}{array}right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="188" HEIGHT="90" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.gif"
 ALT="$\displaystyle \Sigma =  \left(  \begin{array}{*{4}{c}}  c_{11} & c_{12} & \cdot...
...ddots & \vdots    c_{n1} & c_{n2} & \cdots & c_{nn}    \end{array}  \right)$">|; 

$key = q/displaystyle=logsum_QP(Q|X,Theta)p(X|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.gif"
 ALT="$\displaystyle = \log \sum_Q P(Q\vert X,\Theta)p(X\vert\Theta)$">|; 

$key = q/frac{1}{2}left[ldotsright];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.gif"
 ALT="$ \frac{1}{2} \left[\ldots\right]$">|; 

$key = q/displaystyleXinq_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.gif"
 ALT="$\displaystyle X \in q_k$">|; 

$key = q/displaystyleEleft((x_{d1}-mu_{d1})^{mathsfT}(x_{d2}-mu_{d2})right)=0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="200" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.gif"
 ALT="$\displaystyle E\left((x_{d1}-\mu_{d1})^{\mathsf T}(x_{d2}-\mu_{d2})\right) = 0 $">|; 

$key = q/N;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img30.gif"
 ALT="$ N$">|; 

$key = q/displaystylelogp(X_3|ensuremathboldsymbol{Theta}_1),;logp(X_3|ensuremathboldsymbol{Theta}_2),;logp(X_3|ensuremathboldsymbol{Theta}_3),;;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="268" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.gif"
 ALT="$\displaystyle \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_1),\; \log p(X_3\ve...
...th\boldsymbol{\Theta}_2),\; \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_3),\;$">|; 

$key = q/{ensuremathboldsymbol{Sigma}_{text{slashaslash}},ensuremathboldsymbol{Sigma}_{te{text{slashoslash}},ensuremathboldsymbol{Sigma}_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.gif"
 ALT="$ \{\ensuremath\boldsymbol{\Sigma}_{\text{/a/}}, \ensuremath\boldsymbol{\Sigma}_...
...\boldsymbol{\Sigma}_{\text{/o/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/y/}}\}$">|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{k},ensuremathboldsymbol{Sigma}_{k}),;k=1ldotsK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="146" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.gif"
 ALT="$ {\cal N}(\ensuremath\boldsymbol{\mu}_{k},\ensuremath\boldsymbol{\Sigma}_{k}),
\; k=1\ldots K$">|; 

$key = q/xcirclearrowleft{calN}(0,I);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="75" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.gif"
 ALT="$ x \circlearrowleft {\cal N}(0,I)$">|; 

$key = q/X;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img29.gif"
 ALT="$ X$">|; 

$key = q/displaystylemu_{k}^{(i+1)}=frac{sum_{n=1}^{N}x_nP(q_k^{(i)}|x_n,Theta^{(i)})}{sum_{n=1}^{N}P(q_k^{(i)}|x_n,Theta^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.gif"
 ALT="$\displaystyle \mu_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} x_n P(q_k^{(i)}\vert x_n,\Theta^{(i)})}
{\sum_{n=1}^{N} P(q_k^{(i)}\vert x_n,\Theta^{(i)})} $">|; 

$key = q/displaystylehat{ensuremathboldsymbol{mu}}=frac{1}{N}sum_{i=1}^{N}ensuremathmathbf{x}_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="86" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.gif"
 ALT="$ \displaystyle \hat{\ensuremath\boldsymbol{\mu}} = \frac{1}{N}
\sum_{i=1}^{N} \ensuremath\mathbf{x}_i$">|; 

$key = q/[400,1000]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.gif"
 ALT="$ [400,1000]^{\mathsf T}$">|; 

$key = q/displaystyleP(q_k^{(i+1)}|ensuremathboldsymbol{Theta}^{(i+1)})=frac{1}{N}sum_{n=P(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.gif"
 ALT="$\displaystyle P(q_k^{(i+1)}\vert\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac...
...
P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)}) $">|; 

$key = q/hat{mu};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.gif"
 ALT="$ \hat{\mu}$">|; 

$key = q/P(q_k|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.gif"
 ALT="$ P(q_k\vert x)$">|; 

$key = q/displaystylehat{ensuremathboldsymbol{Sigma}}=frac{1}{N-1};sum_{i=1}^{N}(ensuremaymbol{mu})^{mathsfT}(ensuremathmathbf{x}_i-ensuremathboldsymbol{mu});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.gif"
 ALT="$ \displaystyle \hat{\ensuremath\boldsymbol{\Sigma}} =
\frac{1}{N-1} \; \sum_{i...
...dsymbol{\mu})^{\mathsf T}(\ensuremath\mathbf{x}_i-\ensuremath\boldsymbol{\mu}) $">|; 

$key = q/displaystylelogp(x|q_k)+logP(q_k)geqlogp(x|q_l)+logP(q_l);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.gif"
 ALT="$\displaystyle \log p(x\vert q_k)+\log P(q_k) \geq \log p(x\vert q_l)+\log P(q_l)$">|; 

$key = q/displaystyleP(q_k|X,Theta)=frac{p(X|q_k,Theta)P(q_k|Theta)}{p(X|Theta)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.gif"
 ALT="$\displaystyle P(q_k\vert X,\Theta) = \frac{p(X\vert q_k,\Theta) P(q_k\vert\Theta)}{p(X\vert\Theta)}
$">|; 

$key = q/displaystyled_k(x_n),leq,d_l(x_n),;;foralllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.gif"
 ALT="$\displaystyle d_k(x_n)   \leq   d_l(x_n),
\;\; \forall l \neq k
$">|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{text{slashislash}},ensuremathboldsymbol{Sigma}_{text{slashislash}});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.gif"
 ALT="$ {\cal N}(\ensuremath \boldsymbol {\mu }_{\text {/i/}},\ensuremath \boldsymbol {\Sigma }_{\text {/i/}})$">|; 

$key = q/mathbf{A};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img59.gif"
 ALT="$ \mathbf{A}$">|; 

$key = q/displaystyle;;P(q_k|X,Theta)geqP(q_j|X,Theta),;foralljneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="215" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.gif"
 ALT="$\displaystyle \;\; P(q_k\vert X,\Theta) \geq P(q_j\vert X,\Theta), \; \forall j \neq k
$">|; 

$key = q/displaystyleJ=sum_{k=1}^{K}sum_{ensuremathmathbf{x}_ninq_k}d_k(ensuremathmathbf{x}_n);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.gif"
 ALT="$\displaystyle J = \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} d_k(\ensuremath\mathbf{x}_n)
$">|; 

$key = q/Sigma;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img9.gif"
 ALT="$ \Sigma$">|; 

$key = q/mu;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.gif"
 ALT="$ \mu$">|; 

$key = q/displaystylefrac{1}{2}left[-dlogleft(2piright)-logleft(detleft(ensuremathboldsymmbol{Sigma}^{-1}(ensuremathmathbf{x}-ensuremathboldsymbol{mu})right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="331" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.gif"
 ALT="$\displaystyle \frac{1}{2} \left[-d \log \left( 2\pi \right)
- \log \left( \det\...
...dsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})\right]$">|; 

$key = q/displaystylesum_{X}P(X|ensuremathboldsymbol{Theta});=;sum_{Q}sum_{X}p(X,Q|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="212" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.gif"
 ALT="$\displaystyle \sum_{X} P(X\vert\ensuremath\boldsymbol{\Theta}) \;=\; \sum_{Q} \sum_{X} p(X,Q\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyleP(q_k^{(i+1)}|ensuremathboldsymbol{Theta}^{(i+1)})=frac{mbox{numberoiningpointsbelongingto}q_k^{(i)}}{mbox{totalnumberoftrainingpoints}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="387" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.gif"
 ALT="$\displaystyle P(q_k^{(i+1)}\vert\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac...
...ng points
belonging to } q_k^{(i)} }{\mbox{total number of training points}}
$">|; 

$key = q/f_k(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.gif"
 ALT="$ f_k(\ensuremath\mathbf{x}_i)$">|; 

$key = q/k=1ldotsK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img152.gif"
 ALT="$ k=1\ldots K$">|; 

$key = q/{calN}(mu_{k},Sigma_{k}),;k=1ldotsK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.gif"
 ALT="$ {\cal N}(\mu_{k},\Sigma_{k}),
\; k=1\ldots K$">|; 

$key = q/displaystyled_k(ensuremathmathbf{x}_n);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.gif"
 ALT="$\displaystyle d_k(\ensuremath\mathbf{x}_n)$">|; 

$key = q/p(X,Q|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.gif"
 ALT="$ p(X,Q\vert\Theta)$">|; 

$key = q/,slashuslash};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.gif"
 ALT="$ ,/u/\}$">|; 

$key = q/[670,1300]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.gif"
 ALT="$ [670,1300]^{\mathsf T}$">|; 

$key = q/displaystyle,left|ensuremathmathbf{x}_n-ensuremathboldsymbol{mu}_kright|^2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="76" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.gif"
 ALT="$\displaystyle   \left\Vert \ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k \right\Vert^2$">|; 

$key = q/ensuremathmathbf{x}circlearrowleft{calN}(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.gif"
 ALT="$ \ensuremath\mathbf{x}\circlearrowleft {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$">|; 

$key = q/f_{text{slashyslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.gif"
 ALT="$ f_{\text{/y/}}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/1slashK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.gif"
 ALT="$ 1/K$">|; 

$key = q/displaystyle=logsum_{k=1}^{K}P(q_k|X,Theta)p(X|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.gif"
 ALT="$\displaystyle = \log \sum_{k=1}^{K} P(q_k\vert X,\Theta) p(X\vert\Theta)$">|; 

$key = q/displaystyleSigma_1=left[{array}{cc}8000&00&8000{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.gif"
 ALT="$\displaystyle \Sigma_1 = \left[ \begin{array}{cc}
8000 & 0 \\\\
0 & 8000
\end{array} \right]
$">|; 

$key = q/x_i=(F_1,F_2)^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.gif"
 ALT="$ x_i=(F_1,F_2)^{\mathsf T}$">|; 

$key = q/displaystyleP(q_k|ensuremathmathbf{x}_i)geqP(q_l|ensuremathmathbf{x}_i),quadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.gif"
 ALT="$\displaystyle P(q_k\vert\ensuremath\mathbf{x}_i) \geq P(q_l\vert\ensuremath\mathbf{x}_i),\quad \forall l \neq k$">|; 

$key = q/k=1,ldots,K;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.gif"
 ALT="$ k=1,\ldots,K$">|; 

$key = q/ensuremathboldsymbol{Theta}=(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="73" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img63.gif"
 ALT="$ \ensuremath\boldsymbol{\Theta}= (\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$">|; 

$key = q/P(q_k)=1slashK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.gif"
 ALT="$ P(q_k) = 1/K$">|; 

$key = q/k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img130.gif"
 ALT="$ k$">|; 

$key = q/hat{ensuremathboldsymbol{mu}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\mu}}$">|; 

$key = q/X={x_1,x_2,cdots,x_N};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.gif"
 ALT="$ X=\{x_1, x_2,\cdots,x_N\}$">|; 

$key = q/ensuremathboldsymbol{Sigma}_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.gif"
 ALT="$ \ensuremath\boldsymbol{\Sigma}_k$">|; 

$key = q/f_k(ensuremathmathbf{x}_i)=logp(ensuremathmathbf{x}_i|q_k,ensuremathboldsymbol{Theta})+logP(q_k|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="236" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img114.gif"
 ALT="$ f_k(\ensuremath\mathbf{x}_i) = \log
p(\ensuremath\mathbf{x}_i\vert q_k,\ensuremath\boldsymbol{\Theta}) + \log P(q_k\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystylelogP(q_k|X,Theta)simeqlogp(X|q_k,Theta)+logP(q_k|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="282" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.gif"
 ALT="$\displaystyle \log P(q_k\vert X,\Theta) \simeq \log p(X\vert q_k,\Theta) + \log P(q_k\vert\Theta)
$">|; 

$key = q/displaystylexinq_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.gif"
 ALT="$\displaystyle x \in q_k$">|; 

$key = q/P(q_k|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img98.gif"
 ALT="$ P(q_k\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/F_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.gif"
 ALT="$ F_2$">|; 

$key = q/displaystyle=sum_{k=1}^{K}sum_{n=1}^{N}P(q_k|x_n,Theta)logp(x_n|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="207" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.gif"
 ALT="$\displaystyle = \sum_{k=1}^{K} \sum_{n=1}^{N} P(q_k\vert x_n,\Theta) \log p(x_n\vert\Theta)$">|; 

$key = q/hat{ensuremathboldsymbol{Sigma}}_{(10000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(10000)} =$">|; 

$key = q/hat{ensuremathboldsymbol{Sigma}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img51.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\Sigma}}$">|; 

$key = q/includegraphics[height=0.98textheight]{iso};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="748" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img118.gif"
 ALT="\includegraphics[height=0.98\textheight]{iso}">|; 

$key = q/hat{ensuremathboldsymbol{mu}}_{(1000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\mu}}_{(1000)}
=$">|; 

$key = q/x_j;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.gif"
 ALT="$ x_j$">|; 

$key = q/displaystylep(X|Theta)=prod_{i=1}^{N}p(x_i|Theta);;;Leftrightarrow;;;logp(X|Theta)=sum_{i=1}^{N}logp(x_i|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="351" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.gif"
 ALT="$\displaystyle p(X\vert\Theta) = \prod_{i=1}^{N} p(x_i\vert\Theta)
\;\;\; \Leftrightarrow \;\;\;
\log p(X\vert\Theta) = \sum_{i=1}^{N} \log p(x_i\vert\Theta)
$">|; 

$key = q/kin{;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.gif"
 ALT="$ k \in
\{$">|; 

$key = q/P(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.gif"
 ALT="$ P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$">|; 

$key = q/displaystylehat{mu}=frac{1}{N}sum_{i=1}^{N}x_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.gif"
 ALT="$ \displaystyle \hat{\mu} = \frac{1}{N}
\sum_{i=1}^{N} x_i$">|; 

$key = q/logp(X|q_k,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.gif"
 ALT="$ \log p(X\vert q_k,\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystylelogp(x|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.gif"
 ALT="$\displaystyle \log p(x\vert\Theta)$">|; 

$key = q/displaystyle;logp(X_3|Theta_4).;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.gif"
 ALT="$\displaystyle \; \log p(X_3\vert\Theta_4).
$">|; 

$key = q/Sigma_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.gif"
 ALT="$ \Sigma_k$">|; 

$key = q/mu_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.gif"
 ALT="$ \mu_k$">|; 

$key = q/xcirclearrowleft{calN}(mu,Sigma);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.gif"
 ALT="$ x \circlearrowleft {\cal
N}(\mu,\Sigma)$">|; 

$key = q/B;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img52.gif"
 ALT="$ B$">|; 

$key = q/{calN}_1:;ensuremathboldsymbol{Theta}_1=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&00&8000{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="258" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.gif"
 ALT="$ {\cal N}_1: \; \ensuremath\boldsymbol{\Theta}_1 = \left(
\left[\begin{array}{...
...right],
\left[\begin{array}{cc}8000 & 0  0 & 8000\end{array}\right]
\right)$">|; 

$key = q/displaystyleSigma_2=left[{array}{cc}8000&00&18500{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.gif"
 ALT="$\displaystyle \Sigma_2 = \left[ \begin{array}{cc}
8000 & 0 \\\\
0 & 18500
\end{array} \right]
$">|; 

$key = q/displaystyle{calL}(Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.gif"
 ALT="$\displaystyle {\cal L}(\Theta)$">|; 

$key = q/logP(q_{text{slasheslash}}|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.gif"
 ALT="$ \log P(q_{\text{/e/}}\vert x)$">|; 

$key = q/logleft(detleft(Sigmaright)right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="77" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.gif"
 ALT="$ \log \left( \det\left(\Sigma\right) \right)$">|; 

$key = q/hat{mu}_{(10000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.gif"
 ALT="$ \hat{\mu}_{(10000)}
=$">|; 

$key = q/hat{ensuremathboldsymbol{mu}}_{(10000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\mu}}_{(10000)}
=$">|; 

$key = q/ensuremathboldsymbol{Theta}_i=(ensuremathboldsymbol{mu}_i,ensuremathboldsymbol{Sigma}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.gif"
 ALT="$ \ensuremath\boldsymbol{\Theta}_i = (\ensuremath\boldsymbol{\mu}_i,\ensuremath\boldsymbol{\Sigma}_i)$">|; 

$key = q/F_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.gif"
 ALT="$ F_1$">|; 

$key = q/{calN}left(mu_{text{slashislash}},Sigma_{text{slasheslash}}right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.gif"
 ALT="$ {\cal N}\left (\mu _{\text {/i/}},\Sigma _{\text {/e/}}\right )$">|; 

$key = q/Box;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img43.gif"
 ALT="$ \Box$">|; 

$key = q/p(X|q_k,ensuremathboldsymbol{Theta})P(q_k|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.gif"
 ALT="$ p(X\vert q_k,\ensuremath\boldsymbol{\Theta}) P(q_k\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/d-1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.gif"
 ALT="$ d-1$">|; 

$key = q/ensuremathboldsymbol{Sigma};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img10.gif"
 ALT="$ \ensuremath\boldsymbol{\Sigma}$">|; 

$key = q/displaystylep(X|ensuremathboldsymbol{Theta})=prod_{i=1}^{N}p(ensuremathmathbf{x}hboldsymbol{mu},ensuremathboldsymbol{Sigma})}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.gif"
 ALT="$\displaystyle p(X\vert\ensuremath\boldsymbol{\Theta}) =  \prod_{i=1}^{N} p(\ens...
...emath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_3=left[{array}{cc}8000&84008400&18500{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_3 = \left[ \begin{array}{cc}
8000 & 8400 \\\\
8400 & 18500
\end{array} \right]
$">|; 

$key = q/parallelA-Bparallel_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img50.gif"
 ALT="$ \parallel A-B \parallel_2$">|; 

$key = q/displaystylefrac{P(q_k^{(i)}|Theta^{(i)})cdotp(x_n|mu_k^{(i)},Sigma_k^{(i)})}{sum_jP(q_j^{(i)}|Theta^{(i)})cdotp(x_n|mu_j^{(i)},Sigma_j^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.gif"
 ALT="$\displaystyle \frac{P(q_k^{(i)}\vert\Theta^{(i)}) \cdot p(x_n\vert\mu_k^{(i)},\...
...m_j P(q_j^{(i)}\vert\Theta^{(i)}) \cdot p(x_n\vert\mu_j^{(i)},\Sigma_j^{(i)}) }$">|; 

$key = q/x_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.gif"
 ALT="$ x_i$">|; 

$key = q/hat{Sigma}_{(1000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.gif"
 ALT="$ \hat{\Sigma}_{(1000)} =$">|; 

$key = q/ensuremathboldsymbol{mu}_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.gif"
 ALT="$ \ensuremath\boldsymbol{\mu}_k$">|; 

$key = q/displaystyle=logsum_QP(Q|X,ensuremathboldsymbol{Theta})p(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.gif"
 ALT="$\displaystyle = \log \sum_Q P(Q\vert X,\ensuremath\boldsymbol{\Theta})p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyleP(q_k^{(i+1)}|Theta^{(i+1)})=frac{1}{N}sum_{n=1}^{N}P(q_k^{(i)}|x_n,Theta^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="254" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.gif"
 ALT="$\displaystyle P(q_k^{(i+1)}\vert\Theta^{(i+1)}) = \frac{1}{N} \sum_{n=1}^{N} P(q_k^{(i)}\vert x_n,\Theta^{(i)}) $">|; 

$key = q/[P_{text{slashaslash}},P_{text{slasheslash}},P_{text{slashislash}},P_{text{slashoslash}},P_{text{slashyslash}}];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="165" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.gif"
 ALT="$ [P_{\text{/a/}},P_{\text{/e/}},P_{\text{/i/}},P_{\text{/o/}},P_{\text{/y/}}]$">|; 

$key = q/A;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img51.gif"
 ALT="$ A$">|; 

$key = q/displaystylesum_{k=1}^{K}sum_{ensuremathmathbf{x}_ninq_k}logp(ensuremathmathbf{x}_n|ensuremathboldsymbol{Theta}_k),;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="142" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.gif"
 ALT="$\displaystyle \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} \log p(\ensuremath\mathbf{x}_n\vert\ensuremath\boldsymbol{\Theta}_k),$">|; 

$key = q/K;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4.gif"
 ALT="$ K$">|; 

$key = q/mathbf{y}circlearrowleft{calN}(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.gif"
 ALT="$ \mathbf{y} \circlearrowleft {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$">|; 

$key = q/inej;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.gif"
 ALT="$ i\ne j$">|; 

$key = q/{calN}_3:;ensuremathboldsymbol{Theta}_3=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&84008400&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.gif"
 ALT="$ {\cal N}_3: \; \ensuremath\boldsymbol{\Theta}_3 = \left(
\left[\begin{array}{...
...
\left[\begin{array}{cc}8000 & 8400  8400 & 18500\end{array}\right]
\right)$">|; 

$key = q/f_k(ensuremathmathbf{x});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.gif"
 ALT="$ f_k(\ensuremath\mathbf{x})$">|; 

$key = q/i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="8" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img156.gif"
 ALT="$ i$">|; 

$key = q/displaystylep(x|Theta_1)>p(x|Theta_2);;Leftrightarrow;;logp(x|Theta_1)>logp(x|Theta_2);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="306" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img62.gif"
 ALT="$\displaystyle p(x\vert\Theta_1) > p(x\vert\Theta_2) \;\; \Leftrightarrow \;\;
\log p(x\vert\Theta_1) > \log p(x\vert\Theta_2)
$">|; 

$key = q/[420,2500]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.gif"
 ALT="$ [420,2500]^{\mathsf T}$">|; 

$key = q/};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="10" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.gif"
 ALT="$ \}$">|; 

$key = q/(x-mu)^{mathsfT}Sigma^{-1}(x-mu);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.gif"
 ALT="$ (x-\mu)^{\mathsf T}\Sigma^{-1} (x-\mu) $">|; 

$key = q/N=10000;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img32.gif"
 ALT="$ N=10000$">|; 

$key = q/{calN}left(mu_{text{slasheslash}},Sigma_{text{slasheslash}}right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.gif"
 ALT="$ {\cal N}\left (\mu _{\text {/e/}},\Sigma _{\text {/e/}}\right )$">|; 

$key = q/displaystylep(X|ensuremathboldsymbol{Theta})=prod_{i=1}^{N}p(ensuremathmathbf{x}sum_{i=1}^{N}logp(ensuremathmathbf{x}_i|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="470" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.gif"
 ALT="$\displaystyle p(X\vert\ensuremath\boldsymbol{\Theta}) = \prod_{i=1}^{N} p(\ensu...
...{i=1}^{N}
\log p(\ensuremath\mathbf{x}_i\vert\ensuremath\boldsymbol{\Theta})
$">|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{text{slasheslash}},ensuremathboldsymbol{Sigma}_{text{slasheslash}});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="89" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img2.gif"
 ALT="$ {\cal N}(\ensuremath \boldsymbol {\mu }_{\text {/e/}},\ensuremath \boldsymbol {\Sigma }_{\text {/e/}})$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} =$">|; 

$key = q/X={ensuremathmathbf{x}_1,ensuremathmathbf{x}_2,ldots,ensuremathmathbf{x}_N};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.gif"
 ALT="$ X=\{\ensuremath\mathbf{x}_1,
\ensuremath\mathbf{x}_2,\ldots,\ensuremath\mathbf{x}_N\}$">|; 

$key = q/[600,1300]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.gif"
 ALT="$ [600,1300]^{\mathsf T}$">|; 

$key = q/displaystylesum_{k=1}^{K}sum_{x_ninq_k}logp(x_n|Theta_k),;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.gif"
 ALT="$\displaystyle \sum_{k=1}^{K} \sum_{x_n \in q_k} \log p(x_n\vert\Theta_k),$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_1=left[{array}{cc}8000&00&8000{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="138" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}_1 = \left[ \begin{array}{cc}
8000 & 0 \\\\
0 & 8000
\end{array} \right]
$">|; 

$key = q/displaystylec_{ij}=Eleft((x_i-mu_i)^{mathsfT},(x_j-mu_j)right).;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.gif"
 ALT="$\displaystyle c_{ij} = E\left((x_i-\mu_i)^{\mathsf T} (x_j-\mu_j)\right).$">|; 

$key = q/[F_1,F_2];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.gif"
 ALT="$ [F_1,F_2]$">|; 

$key = q/logP(q_{text{slashyslash}}|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img107.gif"
 ALT="$ \log P(q_{\text{/y/}}\vert x)$">|; 

$key = q/displaystylesum_{X}P(X|Theta);=;sum_{Q}sum_{X}p(X,Q|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.gif"
 ALT="$\displaystyle \sum_{X} P(X\vert\Theta) \; = \; \sum_{Q} \sum_{X} p(X,Q\vert\Theta)$">|; 

$key = q/ensuremathmathbf{x}_i=[F_1,F_2]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img112.gif"
 ALT="$ \ensuremath\mathbf{x}_i=[F_1,F_2]^{\mathsf T}$">|; 

$key = q/displaystyleSigma_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.gif"
 ALT="$\displaystyle \Sigma_{k}^{(i+1)} =$">|; 

$key = q/displaystylehat{Sigma}=frac{1}{N-1};sum_{i=1}^{N}(x_i-mu)^{mathsfT}(x_i-mu);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.gif"
 ALT="$ \displaystyle \hat{\Sigma} =
\frac{1}{N-1} \; \sum_{i=1}^{N} (x_i-\mu)^{\mathsf T}(x_i-\mu) $">|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{text{slashislash}},ensuremathboldsymbol{Sigma}_{text{slasheslash}});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.gif"
 ALT="$ {\cal N}(\ensuremath \boldsymbol {\mu }_{\text {/i/}},\ensuremath \boldsymbol {\Sigma }_{\text {/e/}})$">|; 

$key = q/n=1,ldots,N;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.gif"
 ALT="$ n=1,\ldots,N$">|; 

$key = q/displaystyled_k(ensuremathmathbf{x}_n),<,d_l(ensuremathmathbf{x}_n),qquadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.gif"
 ALT="$\displaystyle d_k(\ensuremath\mathbf{x}_n)   <   d_l(\ensuremath\mathbf{x}_n), \qquad \forall l \neq k
$">|; 

$key = q/,;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="7" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.gif"
 ALT="$ ,$">|; 

$key = q/displaystyle(ensuremathmathbf{x}_n-ensuremathboldsymbol{mu}_k)^{mathsfT}(ensuremathmathbf{x}_n-ensuremathboldsymbol{mu}_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.gif"
 ALT="$\displaystyle (\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)^{\mathsf T}(\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)$">|; 

$key = q/P(q_k|X,Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.gif"
 ALT="$ P(q_k\vert X,\Theta)$">|; 

$key = q/displaystyle,left|x_n-mu_kright|^2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.gif"
 ALT="$\displaystyle   \left\Vert x_n - \mu_k \right\Vert^2$">|; 

$key = q/ensuremathboldsymbol{Theta};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img61.gif"
 ALT="$ \ensuremath\boldsymbol{\Theta}$">|; 

$key = q/dlogleft(2piright);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="60" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.gif"
 ALT="$ d\log \left( 2\pi \right)$">|; 

$key = q/displaystyleSigma_{k}^{(i+1)}=frac{sum_{n=1}^{N}P(q_k^{(i)}|x_n,Theta^{(i)})(x_n-mu_k^{(i+1)})^{mathsfT}}{sum_{n=1}^{N}P(q_k^{(i)}|x_n,Theta^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="360" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.gif"
 ALT="$\displaystyle \Sigma_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} P(q_k^{(i)}\vert x_n,\T...
...u_k^{(i+1)})^{\mathsf T}}
{\sum_{n=1}^{N} P(q_k^{(i)}\vert x_n,\Theta^{(i)})} $">|; 

$key = q/ycirclearrowleft{calN}(mu,Sigma);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="78" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.gif"
 ALT="$ y \circlearrowleft {\cal
N}(\mu,\Sigma)$">|; 

$key = q/{calN}_4:;ensuremathboldsymbol{Theta}_4=left(left[{array}{c}2701690{array}right],left[{array}{cc}8000&84008400&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img81.gif"
 ALT="$ {\cal N}_4: \; \ensuremath\boldsymbol{\Theta}_4 = \left(
\left[\begin{array}{...
...
\left[\begin{array}{cc}8000 & 8400  8400 & 18500\end{array}\right]
\right)$">|; 

$key = q/displaystylefrac{1}{sqrt{2pi}^dsqrt{detleft(Sigmaright)}},e^{-frac{1}{2}(x-mu)^{mathsfT}Sigma^{-1}(x-mu)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.gif"
 ALT="$\displaystyle \frac{1}{\sqrt{2\pi}^d \sqrt{\det\left(\Sigma\right)}}
  e^{-\frac{1}{2} (x-\mu)^{\mathsf T}\Sigma^{-1} (x-\mu)}$">|; 

$key = q/displaystylep(x|q_k)P(q_k)geqp(x|q_l)P(q_l);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="172" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.gif"
 ALT="$\displaystyle p(x\vert q_k) P(q_k) \geq p(x\vert q_l) P(q_l)$">|; 

$key = q/displaystyle=logsum_Qp(X,Q|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.gif"
 ALT="$\displaystyle = \log \sum_Q p(X,Q\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/ensuremathmathbf{x}_n;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.gif"
 ALT="$ \ensuremath\mathbf{x}_n$">|; 

$key = q/c_{ij};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.gif"
 ALT="$ c_{ij}$">|; 

$key = q/{calN}_3:;Theta_3=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&84008400&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.gif"
 ALT="$ {\cal N}_3: \; \Theta_3 = \left(
\left[\begin{array}{c}730  1090\end{array}...
...
\left[\begin{array}{cc}8000 & 8400  8400 & 18500\end{array}\right]
\right)$">|; 

$key = q/displaystyleXinq_k;;;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.gif"
 ALT="$\displaystyle X \in q_k \;\;$">|; 

$key = q/displaystyleP(q_k|X,ensuremathboldsymbol{Theta})proptop(X|q_k,ensuremathboldsymbol{Theta});P(q_k|ensuremathboldsymbol{Theta}),quadforallk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="248" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.gif"
 ALT="$\displaystyle P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \propto p(X\vert q_k...
...dsymbol{\Theta})\; P(q_k\vert\ensuremath\boldsymbol{\Theta}), \quad \forall k
$">|; 

$key = q/displaystylemu_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.gif"
 ALT="$\displaystyle \mu_{k}^{(i+1)} =$">|; 

$key = q/x^{mathsfT}Sigmaxge0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.gif"
 ALT="$ x^{\mathsf T}
\Sigma x \ge 0$">|; 

$key = q/ensuremathmathbf{x}^{mathsfT}ensuremathboldsymbol{Sigma},ensuremathmathbf{x}ge0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.gif"
 ALT="$ \ensuremath\mathbf{x}^{\mathsf T}\ensuremath\boldsymbol{\Sigma}  \ensuremath\mathbf{x}\ge 0$">|; 

$key = q/hat{ensuremathboldsymbol{mu}}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="27" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\mu}}_{(100)}
=$">|; 

$key = q/displaystyle=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.gif"
 ALT="$\displaystyle =$">|; 

$key = q/P(q_k|ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.gif"
 ALT="$ P(q_k\vert\ensuremath\mathbf{x}_i)$">|; 

$key = q/{calN}_2:;Theta_2=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&00&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.gif"
 ALT="$ {\cal N}_2: \; \Theta_2 = \left(
\left[\begin{array}{c}730  1090\end{array}\right],
\left[\begin{array}{cc}8000 & 0  0 & 18500\end{array}\right]
\right)$">|; 

$key = q/displaystyleensuremathboldsymbol{mu}_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} =$">|; 

$key = q/I;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img19.gif"
 ALT="$ I$">|; 

$key = q/dtimesd;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.gif"
 ALT="$ d\times d$">|; 

$key = q/ensuremathboldsymbol{mu};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img9.gif"
 ALT="$ \ensuremath\boldsymbol{\mu}$">|; 

$key = q/p(X|q_k,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.gif"
 ALT="$ p(X\vert q_k,\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyle{calL}(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.gif"
 ALT="$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/hat{Sigma}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.gif"
 ALT="$ \hat{\Sigma}_{(100)} =$">|; 

$key = q/{Sigma_{text{slashaslash}},Sigma_{text{slasheslash}},Sigma_{text{slashislash}},Sigma_{text{slashoslash}},Sigma_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.gif"
 ALT="$ \{\Sigma_{\text{/a/}}, \Sigma_{\text{/e/}}, \Sigma_{\text{/i/}},
\Sigma_{\text{/o/}}, \Sigma_{\text{/y/}}\}$">|; 

$key = q/sqrt{Sigma};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.gif"
 ALT="$ \sqrt{\Sigma}$">|; 

$key = q/displaystyle{calL}(Theta)geJ(Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.gif"
 ALT="$\displaystyle {\cal L}(\Theta) \ge J(\Theta)$">|; 

$key = q/hat{ensuremathboldsymbol{Sigma}}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(100)} =$">|; 

$key = q/p(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.gif"
 ALT="$ p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/{calN}(mu,Sigma_3);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.gif"
 ALT="$ {\cal
N}(\mu,\Sigma_3)$">|; 

$key = q/left(logsum_jlambda_jy_jgeqsum_jlambda_jlogy_jmbox{if}sum_jlambda_j=1right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.gif"
 ALT="$ \left( \log \sum_j \lambda_j y_j \geq
\sum_j \lambda_j \log y_j \mbox{ if } \sum_j \lambda_j = 1 \right)$">|; 

$key = q/Ntimes2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.gif"
 ALT="$ N\times2$">|; 

$key = q/displaystylemu=left({array}{c}7301090{array}right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.gif"
 ALT="$\displaystyle \mu = \left( \begin{array}{c} 730  1090 \end{array} \right)
$">|; 

$key = q/displaystylelogp(X_3|Theta_1),;logp(X_3|Theta_2),;logp(X_3|Theta_3),;;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="263" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.gif"
 ALT="$\displaystyle \log p(X_3\vert\Theta_1),\; \log p(X_3\vert\Theta_2),\; \log p(X_3\vert\Theta_3),\;$">|; 

$key = q/displaystyleensuremathmathbf{x}inq_kquadLeftrightarrowquadf_k(ensuremathmathbf{xl(ensuremathmathbf{x},ensuremathboldsymbol{Theta}_l),quadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="281" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.gif"
 ALT="$\displaystyle \ensuremath\mathbf{x}\in q_k \quad \Leftrightarrow \quad f_k(\ens...
...suremath\mathbf{x},\ensuremath\boldsymbol{\Theta}_l),
\quad \forall l \neq k
$">|; 

$key = q/displaystyle=sum_{k=1}^{K}P(q_k|X,Theta)logp(X|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="177" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.gif"
 ALT="$\displaystyle = \sum_{k=1}^{K} P(q_k\vert X,\Theta) \log p(X\vert\Theta)$">|; 

$key = q/2times;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.gif"
 ALT="$ 2\times$">|; 

$key = q/f_{text{slashaslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="52" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.gif"
 ALT="$ f_{\text{/a/}}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/{mu_{text{slashaslash}},mu_{text{slasheslash}},mu_{text{slashislash}},mu_{text{slashoslash}},mu_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.gif"
 ALT="$ \{\mu_{\text{/a/}},
\mu_{\text{/e/}}, \mu_{\text{/i/}}, \mu_{\text{/o/}},
\mu_{\text{/y/}}\}$">|; 

$key = q/displaystyle=logsum_Qp(X,Q|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.gif"
 ALT="$\displaystyle = \log \sum_Q p(X,Q\vert\Theta)$">|; 

$key = q/displaystyleensuremathmathbf{x}inq_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="42" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.gif"
 ALT="$\displaystyle \ensuremath\mathbf{x}\in q_k$">|; 

$key = q/displaystylep(ensuremathmathbf{x}|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.gif"
 ALT="$\displaystyle p(\ensuremath\mathbf{x}\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/displaystyleLeftrightarrow;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.gif"
 ALT="$\displaystyle \Leftrightarrow$">|; 

$key = q/displaystyle=sum_{k=1}^{K}P(q_k|X,ensuremathboldsymbol{Theta})logp(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.gif"
 ALT="$\displaystyle = \sum_{k=1}^{K} P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \log p(X\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/P(q_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.gif"
 ALT="$ P(q_k)$">|; 

$key = q/Theta=(mu,Sigma);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.gif"
 ALT="$ \Theta = (\mu,\Sigma)$">|; 

$key = q/mu_{di}=E(x_{di});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.gif"
 ALT="$ \mu_{di} = E(x_{di})$">|; 

$key = q/c_{ij}=c_{ji};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.gif"
 ALT="$ c_{ij} =
c_{ji}$">|; 

$key = q/[0,1];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.gif"
 ALT="$ [0,1]$">|; 

$key = q/displaystyleq_k^{(i)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.gif"
 ALT="$\displaystyle q_k^{(i)} $">|; 

$key = q/displaystylefrac{1}{2}left[-dlogleft(2piright)-logleft(detleft(Sigmaright)right)-(x-mu)^{mathsfT}Sigma^{-1}(x-mu)right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="325" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.gif"
 ALT="$\displaystyle \frac{1}{2} \left[-d \log \left( 2\pi \right)
- \log \left( \det\left(\Sigma\right) \right)
- (x-\mu)^{\mathsf T}\Sigma^{-1} (x-\mu)\right]$">|; 

$key = q/log(x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.gif"
 ALT="$ \log(x)$">|; 

$key = q/{calL}(Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.gif"
 ALT="$ {\cal
L}(\Theta)$">|; 

$key = q/f_k(x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img109.gif"
 ALT="$ f_k(x)$">|; 

$key = q/Sigma_{text{slasheslash}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.gif"
 ALT="$ \Sigma_{\text{/e/}}$">|; 

$key = q/displaystylexinq_k;;Leftrightarrow;;f_k(x,Theta_k)geqf_l(x,Theta_l),foralllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="255" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.gif"
 ALT="$\displaystyle x \in q_k \;\; \Leftrightarrow \;\; f_k(x,\Theta_k) \geq f_l(x,\Theta_l),
 \forall l \neq k
$">|; 

$key = q/displaystyle;logp(X_3|ensuremathboldsymbol{Theta}_4).;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.gif"
 ALT="$\displaystyle \; \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_4).
$">|; 

$key = q/displaystylep(X|Theta)=prod_{i=1}^{N}p(x_i|Theta)=prod_{i=1}^{N}p(x_i|mu,Sigma)=prod_{i=1}^{N}g_{(mu,Sigma)}(x_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.gif"
 ALT="$\displaystyle p(X\vert\Theta) =
\prod_{i=1}^{N} p(x_i\vert\Theta) =
\prod_{i=1}^{N} p(x_i\vert\mu,\Sigma) =
\prod_{i=1}^{N} g_{(\mu,\Sigma)}(x_i)
$">|; 

$key = q/displaystyleg_{(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma})}(ensurematoldsymbol{Sigma}^{-1}(ensuremathmathbf{x}-ensuremathboldsymbol{mu})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img8.gif"
 ALT="$\displaystyle g_{(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(...
...th\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})}$">|; 

$key = q/displaystyle(x_n-mu_k)^{mathsfT}(x_n-mu_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.gif"
 ALT="$\displaystyle (x_n-\mu_k)^{\mathsf T}(x_n-\mu_k)$">|; 

$key = q/logP(q_{text{slashislash}}|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.gif"
 ALT="$ \log P(q_{\text{/i/}}\vert x)$">|; 

$key = q/J(Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.gif"
 ALT="$ J(\Theta)$">|; 

$key = q/displaystylelogp(ensuremathmathbf{x}|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.gif"
 ALT="$\displaystyle \log p(\ensuremath\mathbf{x}\vert\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/includegraphics[height=0.95textheight]{iso};AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="723" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img138.gif"
 ALT="\includegraphics[height=0.95\textheight]{iso}">|; 

$key = q/hat{Sigma}_{(10000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.gif"
 ALT="$ \hat{\Sigma}_{(10000)} =$">|; 

$key = q/sqrt{ensuremathboldsymbol{Sigma}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.gif"
 ALT="$ \sqrt{\ensuremath\boldsymbol{\Sigma}}$">|; 

$key = q/f_{text{slasheslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.gif"
 ALT="$ f_{\text{/e/}}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/hat{ensuremathboldsymbol{Sigma}}_{(1000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.gif"
 ALT="$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(1000)} =$">|; 

$key = q/hat{mu}_{(1000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.gif"
 ALT="$ \hat{\mu}_{(1000)}
=$">|; 

$key = q/displaystylefrac{P(q_k^{(i)}|Theta^{(i)})cdotp(x_n|q_k^{(i)},Theta^{(i)})}{p(x_n|Theta^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="175" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.gif"
 ALT="$\displaystyle \frac{P(q_k^{(i)}\vert\Theta^{(i)})
\cdot p(x_n\vert q_k^{(i)},\Theta^{(i)})}
{p(x_n\vert\Theta^{(i)})}$">|; 

$key = q/{calN}_4;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.gif"
 ALT="$ {\cal N}_4$">|; 

$key = q/{calN}(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma}_3);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.gif"
 ALT="$ {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma}_3)$">|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}=left[{array}{*{4}{c}}c_{11}&c_{12}&cdotsc_{2n}vdots&vdots&ddots&vdotsc_{n1}&c_{n2}&cdots&c_{nn}{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="183" HEIGHT="90" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.gif"
 ALT="$\displaystyle \ensuremath\boldsymbol{\Sigma}=  \left[  \begin{array}{*{4}{c}}  ...
...ddots & \vdots    c_{n1} & c_{n2} & \cdots & c_{nn}    \end{array}  \right]$">|; 

$key = q/{calL}(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img185.gif"
 ALT="$ {\cal
L}(\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/X_3;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.gif"
 ALT="$ X_3$">|; 

$key = q/{calN}left(mu_{text{slashislash}},Sigma_{text{slashislash}}right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img1.gif"
 ALT="$ {\cal N}\left (\mu _{\text {/i/}},\Sigma _{\text {/i/}}\right )$">|; 

$key = q/logp(ensuremathmathbf{x}_i|q_k,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="90" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.gif"
 ALT="$ \log p(\ensuremath\mathbf{x}_i\vert q_k,\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/ensuremathboldsymbol{Sigma}_3;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.gif"
 ALT="$ \ensuremath\boldsymbol{\Sigma}_3$">|; 

$key = q/displaystyleJ=sum_{k=1}^{K}sum_{x_ninq_k}d_k(x_n);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="58" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.gif"
 ALT="$\displaystyle J = \sum_{k=1}^{K} \sum_{x_n \in q_k} d_k(x_n)
$">|; 

$key = q/displaystylelogp(ensuremathmathbf{x}_i|q_k)+logP(q_k)geqlogp(ensuremathmathbf{x}_i|q_l)+logP(q_l),quadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="357" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.gif"
 ALT="$\displaystyle \log p(\ensuremath\mathbf{x}_i\vert q_k)+\log P(q_k) \geq \log
p(\ensuremath\mathbf{x}_i\vert q_l)+\log P(q_l),\quad \forall l \neq k$">|; 

$key = q/Q;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.gif"
 ALT="$ Q$">|; 

$key = q/c_{ii};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.gif"
 ALT="$ c_{ii}$">|; 

$key = q/(ensuremathmathbf{x}-ensuremathboldsymbol{mu})^{mathsfT}ensuremathboldsymbol{Sigma}^{-1}(ensuremathmathbf{x}-ensuremathboldsymbol{mu});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img76.gif"
 ALT="$ (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})^{\mathsf T}\ensuremath\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})$">|; 

$key = q/f_{text{slashislash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.gif"
 ALT="$ f_{\text{/i/}}(\ensuremath\mathbf{x}_i)$">|; 

$key = q/displaystyle{calL}(ensuremathboldsymbol{Theta})geJ(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="87" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.gif"
 ALT="$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) \ge J(\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/c_{ij}=0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.gif"
 ALT="$ c_{ij} = 0$">|; 

$key = q/mathbf{B};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img60.gif"
 ALT="$ \mathbf{B}$">|; 

$key = q/x_n;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.gif"
 ALT="$ x_n$">|; 

$key = q/displaystylep(ensuremathmathbf{x}_i|q_k);P(q_k)geqp(ensuremathmathbf{x}_i|q_l);P(q_l),quadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="249" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.gif"
 ALT="$\displaystyle p(\ensuremath\mathbf{x}_i\vert q_k)\; P(q_k) \geq p(\ensuremath\mathbf{x}_i\vert q_l)\; P(q_l),\quad
\forall l \neq k$">|; 

$key = q/mathbf{I};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="9" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img26.gif"
 ALT="$ \mathbf{I}$">|; 

$key = q/Sigma_3;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.gif"
 ALT="$ \Sigma_3$">|; 

$key = q/displaystylep(x|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="44" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.gif"
 ALT="$\displaystyle p(x\vert\Theta)$">|; 

$key = q/hat{Sigma};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img43.gif"
 ALT="$ \hat{\Sigma}$">|; 

$key = q/displaystylefrac{1}{sqrt{2pi}^dsqrt{detleft(ensuremathboldsymbol{Sigma}right)}},oldsymbol{Sigma}^{-1}(ensuremathmathbf{x}-ensuremathboldsymbol{mu})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="217" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.gif"
 ALT="$\displaystyle \frac{1}{\sqrt{2\pi}^d \sqrt{\det\left(\ensuremath\boldsymbol{\Si...
...th\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})}$">|; 

$key = q/displaystyleSigma_3=left[{array}{cc}8000&84008400&18500{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="144" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.gif"
 ALT="$\displaystyle \Sigma_3 = \left[ \begin{array}{cc}
8000 & 8400 \\\\
8400 & 18500
\end{array} \right]
$">|; 

$key = q/displaystyleP(q_k^{(i+1)}|Theta^{(i+1)})=frac{mbox{numberoftrainingpointsbelongingto}q_k^{(i)}}{mbox{totalnumberoftrainingpoints}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="386" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.gif"
 ALT="$\displaystyle P(q_k^{(i+1)}\vert\Theta^{(i+1)}) = \frac{\mbox{number of training points
belonging to } q_k^{(i)} }{\mbox{total number of training points}}$">|; 

$key = q/{calN}_3;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.gif"
 ALT="$ {\cal N}_3$">|; 

$key = q/J(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img184.gif"
 ALT="$ J(\ensuremath\boldsymbol{\Theta})$">|; 

$key = q/X_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.gif"
 ALT="$ X_2$">|; 

$key = q/log;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.gif"
 ALT="$ \log$">|; 

$key = q/logP(q_{text{slashaslash}}|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img103.gif"
 ALT="$ \log P(q_{\text{/a/}}\vert x)$">|; 

$key = q/logP(q_{text{slashoslash}}|x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.gif"
 ALT="$ \log P(q_{\text{/o/}}\vert x)$">|; 

$key = q/ensuremathboldsymbol{Sigma}_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.gif"
 ALT="$ \ensuremath\boldsymbol{\Sigma}_2$">|; 

$key = q/(ensuremathboldsymbol{mu}_k,ensuremathboldsymbol{Sigma}_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.gif"
 ALT="$ (\ensuremath\boldsymbol{\mu}_k,\ensuremath\boldsymbol{\Sigma}_k)$">|; 

$key = q/displaystyleP(q_k^{(i)}|x_n,Theta^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.gif"
 ALT="$\displaystyle P(q_k^{(i)}\vert x_n,\Theta^{(i)})$">|; 

$key = q/{calN}_4:;Theta_4=left(left[{array}{c}2701690{array}right],left[{array}{cc}8000&84008400&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="264" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.gif"
 ALT="$ {\cal N}_4: \; \Theta_4 = \left(
\left[\begin{array}{c}270  1690\end{array}...
...
\left[\begin{array}{cc}8000 & 8400  8400 & 18500\end{array}\right]
\right)$">|; 

$key = q/mathbf{y}=ensuremathboldsymbol{mu}+sqrt{ensuremathboldsymbol{Sigma}},ensuremathmathbf{x};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.gif"
 ALT="$ \mathbf{y} = \ensuremath\boldsymbol{\mu}+
\sqrt{\ensuremath\boldsymbol{\Sigma}} \ensuremath\mathbf{x}$">|; 

$key = q/displaystylefrac{P(q_k^{(i)}|ensuremathboldsymbol{Theta}^{(i)})cdotp(ensuremathmsuremathboldsymbol{mu}_j^{(i)},ensuremathboldsymbol{Sigma}_j^{(i)})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="55" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img171.gif"
 ALT="$\displaystyle \frac{P(q_k^{(i)}\vert\ensuremath\boldsymbol{\Theta}^{(i)}) \cdot...
...rt\ensuremath\boldsymbol{\mu}_j^{(i)},\ensuremath\boldsymbol{\Sigma}_j^{(i)}) }$">|; 

$key = q/E(x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.gif"
 ALT="$ E(x)$">|; 

$key = q/d;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img5.gif"
 ALT="$ d$">|; 

$key = q/P(q_k|Theta);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="28" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.gif"
 ALT="$ P(q_k\vert\Theta)$">|; 

$key = q/x;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img13.gif"
 ALT="$ x$">|; 

$key = q/{calN}_2:;ensuremathboldsymbol{Theta}_2=left(left[{array}{c}7301090{array}right],left[{array}{cc}8000&00&18500{array}right]right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="265" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.gif"
 ALT="$ {\cal N}_2: \; \ensuremath\boldsymbol{\Theta}_2 = \left(
\left[\begin{array}{...
...ight],
\left[\begin{array}{cc}8000 & 0  0 & 18500\end{array}\right]
\right)$">|; 

$key = q/x_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.gif"
 ALT="$ x_1$">|; 

$key = q/k^{text{th}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img143.gif"
 ALT="$ k^{\text{th}}$">|; 

$key = q/hat{mu}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="26" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img48.gif"
 ALT="$ \hat{\mu}_{(100)}
=$">|; 

$key = q/circlearrowleft;MSF=1.4;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="25" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.gif"
 ALT="$ \circlearrowleft$">|; 

1;