www.pudn.com > Gaussiangaijin.zip > images.pl


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


$key = q/displaystyleensuremathboldsymbol{Sigma}_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} =$|; 

$key = q/ensuremathmathbf{x}circlearrowleft{calN}(mathbf{0},mathbf{I});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \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|$ \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|$\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|$\displaystyle \frac{P(q_k^{(i)}\vert\ensuremath\boldsymbol{\Theta}^{(i)})
\cdot...
...}^{(i)})}
{p(\ensuremath\mathbf{x}_n\vert\ensuremath\boldsymbol{\Theta}^{(i)})}$|; 

$key = q/[600,1300]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ [600,1300]^{\mathsf T}$|; 

$key = q/X={ensuremathmathbf{x}_1,ensuremathmathbf{x}_2,ldots,ensuremathmathbf{x}_N};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ X=\{\ensuremath\mathbf{x}_1,
\ensuremath\mathbf{x}_2,\ldots,\ensuremath\mathbf{x}_N\}$|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_1=left[{array}{cc}8000&00&8000{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \ensuremath\boldsymbol{\Sigma}_1 = \left[ \begin{array}{cc}
8000 & 0 \\\\
0 & 8000
\end{array} \right]
$|; 

$key = q/p(ensuremathmathbf{x}_i|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ p(\ensuremath\mathbf{x}_i\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/displaystylec_{ij}=Eleft((x_i-mu_i)^{mathsfT},(x_j-mu_j)right).;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle c_{ij} = E\left((x_i-\mu_i)^{\mathsf T} (x_j-\mu_j)\right).$|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_2=left[{array}{cc}8000&00&18500{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \ensuremath\boldsymbol{\Sigma}_2 = \left[ \begin{array}{cc}
8000 & 0 \\\\
0 & 18500
\end{array} \right]
$|; 

$key = q/[F_1,F_2];MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ [F_1,F_2]$|; 

$key = q/f_{text{slashoslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ f_{\text{/o/}}(\ensuremath\mathbf{x}_i)$|; 

$key = q/ensuremathmathbf{x}_i=[F_1,F_2]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\mathbf{x}_i=[F_1,F_2]^{\mathsf T}$|; 

$key = q/displaystyleP(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$|; 

$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|$\displaystyle \log P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \propto \log p(...
...ensuremath\boldsymbol{\Theta}) + \log P(q_k\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{text{slashislash}},ensuremathboldsymbol{Sigma}_{text{slasheslash}});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ {\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|$ 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|$\displaystyle d_k(\ensuremath\mathbf{x}_n)   <   d_l(\ensuremath\mathbf{x}_n), \qquad \forall l \neq k
$|; 

$key = q/p(X,Q|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ p(X,Q\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/{calN}_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ {\cal N}_2$|; 

$key = q/P(q_k|X,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ P(q_k\vert X,\ensuremath\boldsymbol{\Theta})$|; 

$key = q/[530,1000]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ [530,1000]^{\mathsf T}$|; 

$key = q/,;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ ,$|; 

$key = q/ensuremathboldsymbol{Sigma}_{text{slasheslash}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}$|; 

$key = q/X_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ X_1$|; 

$key = q/|mathbf{A}-mathbf{B}|_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \Vert\mathbf{A}-\mathbf{B}\Vert _2$|; 

$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|$\displaystyle (\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)^{\mathsf T}(\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)$|; 

$key = q/ensuremathboldsymbol{Theta};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\boldsymbol{\Theta}$|; 

$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|$\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/dlogleft(2piright);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ d\log \left( 2\pi \right)$|; 

$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|$ \{\ensuremath\boldsymbol{\mu}_{\text{/a/}},
\ensuremath\boldsymbol{\mu}_{\tex...
...emath\boldsymbol{\mu}_{\text{/o/}},
\ensuremath\boldsymbol{\mu}_{\text{/y/}}\}$|; 

$key = q/displaystylequadP(q_k|X,ensuremathboldsymbol{Theta})geqP(q_j|X,ensuremathboldsymbol{Theta}),quadforalljneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\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/{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|$ {\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/q_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ 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|$\displaystyle = \sum_{k=1}^{K} \sum_{n=1}^{N} P(q_k\vert\ensuremath\mathbf{x}_n...
...bol{\Theta}) \log p(\ensuremath\mathbf{x}_n\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/ensuremathmathbf{x}_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\mathbf{x}_i$|; 

$key = q/displaystyle=logsum_Qp(X,Q|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle = \log \sum_Q p(X,Q\vert\ensuremath\boldsymbol{\Theta})$|; 

$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|$\displaystyle P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) = \frac{p(X\vert q_k...
...k\vert\ensuremath\boldsymbol{\Theta})}{p(X\vert\ensuremath\boldsymbol{\Theta})}$|; 

$key = q/ensuremathmathbf{x}_n;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\mathbf{x}_n$|; 

$key = q/[400,1800]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ [400,1800]^{\mathsf T}$|; 

$key = q/c_{ij};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ c_{ij}$|; 

$key = q/displaystyle{calL}(ensuremathboldsymbol{Theta})=logp(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) = \log p(X\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/p(X|ensuremathboldsymbol{Theta})=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ p(X\vert\ensuremath\boldsymbol{\Theta})=$|; 

$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|$\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/displaystyleensuremathboldsymbol{mu}=left[{array}{c}7301090{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\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|$\displaystyle = \log \sum_{k=1}^{K} P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) p(X\vert\ensuremath\boldsymbol{\Theta})$|; 

$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|$\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|$ q_k^{(i)}$|; 

$key = q/mu_i=E(x_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \mu_i = E(x_i)$|; 

$key = q/logleft(detleft(ensuremathboldsymbol{Sigma}right)right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \log \left( \det\left(\ensuremath\boldsymbol{\Sigma}\right)
\right)$|; 

$key = q/ensuremathmathbf{x};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\mathbf{x}$|; 

$key = q/ensuremathmathbf{x}^{mathsfT}ensuremathboldsymbol{Sigma},ensuremathmathbf{x}ge0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\mathbf{x}^{\mathsf T}\ensuremath\boldsymbol{\Sigma}  \ensuremath\mathbf{x}\ge 0$|; 

$key = q/{calN}_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ {\cal N}_1$|; 

$key = q/hat{ensuremathboldsymbol{mu}}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \hat{\ensuremath\boldsymbol{\mu}}_{(100)}
=$|; 

$key = q/displaystyle=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle =$|; 

$key = q/P(q_k|ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ P(q_k\vert\ensuremath\mathbf{x}_i)$|; 

$key = q/frac{1}{2}left[ldotsright];MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \frac{1}{2}
\left[\ldots\right]$|; 

$key = q/displaystyleensuremathboldsymbol{mu}_{k}^{(i+1)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} =$|; 

$key = q/displaystyleXinq_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle X \in q_k$|; 

$key = q/N;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ 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|$\displaystyle \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_1),\; \log p(X_3\ve...
...th\boldsymbol{\Theta}_2),\; \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_3),\;$|; 

$key = q/{ensuremathboldsymbol{Sigma}_{text{slashaslash}},ensuremathboldsymbol{Sigma}_{te{text{slashoslash}},ensuremathboldsymbol{Sigma}_{text{slashyslash}}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \{\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|$ {\cal N}(\ensuremath\boldsymbol{\mu}_{k},\ensuremath\boldsymbol{\Sigma}_{k}),
\; k=1\ldots K$|; 

$key = q/X;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ X$|; 

$key = q/displaystylehat{ensuremathboldsymbol{mu}}=frac{1}{N}sum_{i=1}^{N}ensuremathmathbf{x}_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \displaystyle \hat{\ensuremath\boldsymbol{\mu}} = \frac{1}{N}
\sum_{i=1}^{N} \ensuremath\mathbf{x}_i$|; 

$key = q/ensuremathboldsymbol{mu};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\boldsymbol{\mu}$|; 

$key = q/dtimesd;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ d\times d$|; 

$key = q/p(X|q_k,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ p(X\vert q_k,\ensuremath\boldsymbol{\Theta})$|; 

$key = q/displaystyle{calL}(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta})$|; 

$key = q/[400,1000]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ [400,1000]^{\mathsf T}$|; 

$key = q/hat{ensuremathboldsymbol{Sigma}}_{(100)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(100)} =$|; 

$key = q/p(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ p(X\vert\ensuremath\boldsymbol{\Theta})$|; 

$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|$\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/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|$ \displaystyle \hat{\ensuremath\boldsymbol{\Sigma}} =
\frac{1}{N-1} \; \sum_{i...
...dsymbol{\mu})^{\mathsf T}(\ensuremath\mathbf{x}_i-\ensuremath\boldsymbol{\mu}) $|; 

$key = q/{calN}(ensuremathboldsymbol{mu}_{text{slashislash}},ensuremathboldsymbol{Sigma}_{text{slashislash}});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ {\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|$ \mathbf{A}$|; 

$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|$\displaystyle J = \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} d_k(\ensuremath\mathbf{x}_n)
$|; 

$key = q/left(logsum_jlambda_jy_jgeqsum_jlambda_jlogy_jmbox{if}sum_jlambda_j=1right);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \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|$ N\times2$|; 

$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|$\displaystyle \frac{1}{2} \left[-d \log \left( 2\pi \right)
- \log \left( \det\...
...dsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})\right]$|; 

$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|$\displaystyle \sum_{X} P(X\vert\ensuremath\boldsymbol{\Theta}) \;=\; \sum_{Q} \sum_{X} p(X,Q\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/displaystyleensuremathmathbf{x}inq_kquadLeftrightarrowquadf_k(ensuremathmathbf{xl(ensuremathmathbf{x},ensuremathboldsymbol{Theta}_l),quadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\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/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|$\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|$ f_k(\ensuremath\mathbf{x}_i)$|; 

$key = q/k=1ldotsK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ k=1\ldots K$|; 

$key = q/f_{text{slashaslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ f_{\text{/a/}}(\ensuremath\mathbf{x}_i)$|; 

$key = q/displaystyled_k(ensuremathmathbf{x}_n);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle d_k(\ensuremath\mathbf{x}_n)$|; 

$key = q/2times;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ 2\times$|; 

$key = q/,slashuslash};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ ,/u/\}$|; 

$key = q/displaystyleensuremathmathbf{x}inq_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \ensuremath\mathbf{x}\in q_k$|; 

$key = q/displaystylep(ensuremathmathbf{x}|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle p(\ensuremath\mathbf{x}\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/[670,1300]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ [670,1300]^{\mathsf T}$|; 

$key = q/displaystyle,left|ensuremathmathbf{x}_n-ensuremathboldsymbol{mu}_kright|^2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle   \left\Vert \ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k \right\Vert^2$|; 

$key = q/ensuremathmathbf{x}circlearrowleft{calN}(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \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|$ f_{\text{/y/}}(\ensuremath\mathbf{x}_i)$|; 

$key = q/displaystyleLeftrightarrow;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \Leftrightarrow$|; 

$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|$\displaystyle = \sum_{k=1}^{K} P(q_k\vert X,\ensuremath\boldsymbol{\Theta}) \log p(X\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/1slashK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ 1/K$|; 

$key = q/P(q_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ P(q_k)$|; 

$key = q/k=1,ldots,K;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ k=1,\ldots,K$|; 

$key = q/displaystyleP(q_k|ensuremathmathbf{x}_i)geqP(q_l|ensuremathmathbf{x}_i),quadforalllneqk;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle P(q_k\vert\ensuremath\mathbf{x}_i) \geq P(q_l\vert\ensuremath\mathbf{x}_i),\quad \forall l \neq k$|; 

$key = q/c_{ij}=c_{ji};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ c_{ij} = c_{ji}$|; 

$key = q/ensuremathboldsymbol{Theta}=(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\boldsymbol{\Theta}= (\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$|; 

$key = q/[0,1];MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ [0,1]$|; 

$key = q/P(q_k)=1slashK;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ P(q_k) = 1/K$|; 

$key = q/displaystyleq_k^{(i)};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle q_k^{(i)}
$|; 

$key = q/k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ k$|; 

$key = q/hat{ensuremathboldsymbol{mu}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \hat{\ensuremath\boldsymbol{\mu}}$|; 

$key = q/log(x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \log(x)$|; 

$key = q/ensuremathboldsymbol{Sigma}_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \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|$ 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/displaystyle;logp(X_3|ensuremathboldsymbol{Theta}_4).;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \; \log p(X_3\vert\ensuremath\boldsymbol{\Theta}_4).
$|; 

$key = q/P(q_k|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ P(q_k\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/displaystyleg_{(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma})}(ensurematoldsymbol{Sigma}^{-1}(ensuremathmathbf{x}-ensuremathboldsymbol{mu})};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle g_{(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(...
...th\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})}$|; 

$key = q/hat{ensuremathboldsymbol{Sigma}}_{(10000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(10000)} =$|; 

$key = q/hat{ensuremathboldsymbol{Sigma}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \hat{\ensuremath\boldsymbol{\Sigma}}$|; 

$key = q/displaystylelogp(ensuremathmathbf{x}|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \log p(\ensuremath\mathbf{x}\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/includegraphics[height=0.95textheight]{iso};AAT/;
$cached_env_img{$key} = q|\includegraphics[height=0.95\textheight]{iso}|; 

$key = q/sqrt{ensuremathboldsymbol{Sigma}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \sqrt{\ensuremath\boldsymbol{\Sigma}}$|; 

$key = q/x_j;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ x_j$|; 

$key = q/hat{ensuremathboldsymbol{mu}}_{(1000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \hat{\ensuremath\boldsymbol{\mu}}_{(1000)}
=$|; 

$key = q/f_{text{slasheslash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ f_{\text{/e/}}(\ensuremath\mathbf{x}_i)$|; 

$key = q/kin{;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ k \in
\{$|; 

$key = q/P(q_k^{(i)}|ensuremathmathbf{x}_n,ensuremathboldsymbol{Theta}^{(i)});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ P(q_k^{(i)}\vert\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$|; 

$key = q/hat{ensuremathboldsymbol{Sigma}}_{(1000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \hat{\ensuremath\boldsymbol{\Sigma}}_{(1000)} =$|; 

$key = q/logp(X|q_k,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \log p(X\vert q_k,\ensuremath\boldsymbol{\Theta})$|; 

$key = q/{calN}_4;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ {\cal N}_4$|; 

$key = q/{calN}(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma}_3);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ {\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|$\displaystyle \ensuremath\boldsymbol{\Sigma}=  \left[  \begin{array}{*{4}{c}}  ...
...ddots & \vdots \   c_{n1} & c_{n2} & \cdots & c_{nn} \   \end{array}  \right]$|; 

$key = q/{calL}(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ {\cal
L}(\ensuremath\boldsymbol{\Theta})$|; 

$key = q/X_3;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ X_3$|; 

$key = q/logp(ensuremathmathbf{x}_i|q_k,ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \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|$ \ensuremath\boldsymbol{\Sigma}_3$|; 

$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|$ {\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/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|$\displaystyle \log p(\ensuremath\mathbf{x}_i\vert q_k)+\log P(q_k) \geq \log
p(\ensuremath\mathbf{x}_i\vert q_l)+\log P(q_l),\quad \forall l \neq k$|; 

$key = q/Q;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ Q$|; 

$key = q/c_{ii};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ 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|$ (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})^{\mathsf T}\ensuremath\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})$|; 

$key = q/ensuremathboldsymbol{Theta}_i=(ensuremathboldsymbol{mu}_i,ensuremathboldsymbol{Sigma}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\boldsymbol{\Theta}_i = (\ensuremath\boldsymbol{\mu}_i,\ensuremath\boldsymbol{\Sigma}_i)$|; 

$key = q/hat{ensuremathboldsymbol{mu}}_{(10000)}=;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \hat{\ensuremath\boldsymbol{\mu}}_{(10000)}
=$|; 

$key = q/Box;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \Box$|; 

$key = q/f_{text{slashislash}}(ensuremathmathbf{x}_i);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ f_{\text{/i/}}(\ensuremath\mathbf{x}_i)$|; 

$key = q/p(X|q_k,ensuremathboldsymbol{Theta})P(q_k|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ 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|$ d-1$|; 

$key = q/ensuremathboldsymbol{Sigma};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \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|$\displaystyle p(X\vert\ensuremath\boldsymbol{\Theta}) =  \prod_{i=1}^{N} p(\ens...
...emath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(\ensuremath\mathbf{x}_i)$|; 

$key = q/displaystyleensuremathboldsymbol{Sigma}_3=left[{array}{cc}8000&84008400&18500{array}right];MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle \ensuremath\boldsymbol{\Sigma}_3 = \left[ \begin{array}{cc}
8000 & 8400 \\\\
8400 & 18500
\end{array} \right]
$|; 

$key = q/displaystyle{calL}(ensuremathboldsymbol{Theta})geJ(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) \ge J(\ensuremath\boldsymbol{\Theta})$|; 

$key = q/c_{ij}=0;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ c_{ij} = 0$|; 

$key = q/x_i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ x_i$|; 

$key = q/mathbf{B};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \mathbf{B}$|; 

$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|$\displaystyle p(\ensuremath\mathbf{x}_i\vert q_k)\; P(q_k) \geq p(\ensuremath\mathbf{x}_i\vert q_l)\; P(q_l),\quad
\forall l \neq k$|; 

$key = q/mathbf{I};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \mathbf{I}$|; 

$key = q/ensuremathboldsymbol{mu}_k;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\boldsymbol{\mu}_k$|; 

$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|$\displaystyle \frac{1}{\sqrt{2\pi}^d \sqrt{\det\left(\ensuremath\boldsymbol{\Si...
...th\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})}$|; 

$key = q/displaystyle=logsum_QP(Q|X,ensuremathboldsymbol{Theta})p(X|ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$\displaystyle = \log \sum_Q P(Q\vert X,\ensuremath\boldsymbol{\Theta})p(X\vert\ensuremath\boldsymbol{\Theta})$|; 

$key = q/{calN}_3;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ {\cal
N}_3$|; 

$key = q/J(ensuremathboldsymbol{Theta});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ J(\ensuremath\boldsymbol{\Theta})$|; 

$key = q/log;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \log$|; 

$key = q/X_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ X_2$|; 

$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|$ [P_{\text{/a/}},P_{\text{/e/}},P_{\text{/i/}},P_{\text{/o/}},P_{\text{/y/}}]$|; 

$key = q/ensuremathboldsymbol{Sigma}_2;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \ensuremath\boldsymbol{\Sigma}_2$|; 

$key = q/(ensuremathboldsymbol{mu}_k,ensuremathboldsymbol{Sigma}_k);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ (\ensuremath\boldsymbol{\mu}_k,\ensuremath\boldsymbol{\Sigma}_k)$|; 

$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|$\displaystyle \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} \log p(\ensuremath\mathbf{x}_n\vert\ensuremath\boldsymbol{\Theta}_k),$|; 

$key = q/K;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ K$|; 

$key = q/mathbf{y}circlearrowleft{calN}(ensuremathboldsymbol{mu},ensuremathboldsymbol{Sigma});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \mathbf{y} \circlearrowleft {\cal
N}(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})$|; 

$key = q/mathbf{y}=ensuremathboldsymbol{mu}+sqrt{ensuremathboldsymbol{Sigma}},ensuremathmathbf{x};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \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|$\displaystyle \frac{P(q_k^{(i)}\vert\ensuremath\boldsymbol{\Theta}^{(i)}) \cdot...
...rt\ensuremath\boldsymbol{\mu}_j^{(i)},\ensuremath\boldsymbol{\Sigma}_j^{(i)}) }$|; 

$key = q/E(x);MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ E(x)$|; 

$key = q/f_k(ensuremathmathbf{x});MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ f_k(\ensuremath\mathbf{x})$|; 

$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|$ {\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/inej;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ i\ne j$|; 

$key = q/d;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ d$|; 

$key = q/i;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ i$|; 

$key = q/[420,2500]^{mathsfT};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ [420,2500]^{\mathsf T}$|; 

$key = q/x;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ x$|; 

$key = q/};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \}$|; 

$key = q/x_1;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ x_1$|; 

$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|$ {\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/N=10000;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ N=10000$|; 

$key = q/k^{text{th}};MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ k^{\text{th}}$|; 

$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|$\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|$ {\cal N}(\ensuremath \boldsymbol {\mu }_{\text {/e/}},\ensuremath \boldsymbol {\Sigma }_{\text {/e/}})$|; 

$key = q/circlearrowleft;MSF=1.4;AAT/;
$cached_env_img{$key} = q|$ \circlearrowleft$|; 

1;