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A Tutorial for the Course Computational Intelligence\end{rawhtml}
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% !!! IMAGES START HERE !!!
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$\displaystyle g_{(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(\ensuremath\mathbf{x}) = \frac{1}{\sqrt{2\pi}^d \sqrt{\det\left(\ensuremath\boldsymbol{\Sigma}\right)}} \, e^{-\frac{1}{2} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})^{\mathsf T} \ensuremath\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})}$%
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$\displaystyle \ensuremath\boldsymbol{\Sigma}= \left[ \begin{array}{*{4}{c}} c_{11} & c_{12} & \cdots & c_{1n} \\ c_{21} & c_{22} & \cdots & c_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ c_{n1} & c_{n2} & \cdots & c_{nn} \\ \end{array} \right]$%
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$ \sqrt{\ensuremath\boldsymbol{\Sigma}}$%
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$ \mathbf{I}$%
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$ \mathbf{y} = \ensuremath\boldsymbol{\mu}+
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$ N=10000$%
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$\displaystyle \ensuremath\boldsymbol{\mu}= \left[ \begin{array}{c} 730 \\1090 \end{array} \right]
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$ X_1$%
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$ \hat{\ensuremath\boldsymbol{\mu}}$%
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$ \hat{\ensuremath\boldsymbol{\mu}}_{(10000)}
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$ \hat{\ensuremath\boldsymbol{\mu}}_{(1000)}
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$ \ensuremath\boldsymbol{\Theta}$%
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$\displaystyle p(X|\ensuremath\boldsymbol{\Theta}) = \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta}) = \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma}) = \prod_{i=1}^{N} g_{(\ensuremath\boldsymbol{\mu},\ensuremath\boldsymbol{\Sigma})}(\ensuremath\mathbf{x}_i)$%
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$\displaystyle p(X|\ensuremath\boldsymbol{\Theta}) = \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta}) \quad \Leftrightarrow \quad
\log p(X|\ensuremath\boldsymbol{\Theta}) = \log \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta}) = \sum_{i=1}^{N}
\log p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta})
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$\displaystyle p(\ensuremath\mathbf{x}|\ensuremath\boldsymbol{\Theta})$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay3487}%
$\displaystyle =$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3489}%
$\displaystyle \frac{1}{\sqrt{2\pi}^d \sqrt{\det\left(\ensuremath\boldsymbol{\Sigma}\right)}}
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$\displaystyle \log p(\ensuremath\mathbf{x}|\ensuremath\boldsymbol{\Theta})$%
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$\displaystyle \frac{1}{2} \left[-d \log \left( 2\pi \right)
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$ \log(x)$%
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\lthtmlinlinemathA{tex2html_wrap_indisplay3499}%
$\displaystyle p(x|\ensuremath\boldsymbol{\Theta}_1) > p(x|\ensuremath\boldsymbol{\Theta}_2) \quad \Leftrightarrow \quad
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$%
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{\newpage\clearpage
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$ \frac{1}{2}
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$ d\log \left( 2\pi \right)$%
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$ \log \left( \det\left(\ensuremath\boldsymbol{\Sigma}\right)
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3521}%
$ (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})^{\mathsf T}\ensuremath\boldsymbol{\Sigma}^{-1} (\ensuremath\mathbf{x}-\ensuremath\boldsymbol{\mu})$%
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$ \ensuremath\boldsymbol{\Theta}_i = (\ensuremath\boldsymbol{\mu}_i,\ensuremath\boldsymbol{\Sigma}_i)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3549}%
$ {\cal N}_1: \; \ensuremath\boldsymbol{\Theta}_1 = \left(
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\left[\begin{array}{cc}8000 & 0 \\0 & 8000\end{array}\right]
\right)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3551}%
$ {\cal N}_2: \; \ensuremath\boldsymbol{\Theta}_2 = \left(
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\left[\begin{array}{cc}8000 & 0 \\0 & 18500\end{array}\right]
\right)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3553}%
$ {\cal N}_3: \; \ensuremath\boldsymbol{\Theta}_3 = \left(
\left[\begin{array}{c}730 \\1090\end{array}\right],
\left[\begin{array}{cc}8000 & 8400 \\8400 & 18500\end{array}\right]
\right)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3555}%
$ {\cal N}_4: \; \ensuremath\boldsymbol{\Theta}_4 = \left(
\left[\begin{array}{c}270 \\1690\end{array}\right],
\left[\begin{array}{cc}8000 & 8400 \\8400 & 18500\end{array}\right]
\right)$%
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$\displaystyle \log p(X_3|\ensuremath\boldsymbol{\Theta}_1),\; \log p(X_3|\ensuremath\boldsymbol{\Theta}_2),\; \log p(X_3|\ensuremath\boldsymbol{\Theta}_3),\;$%
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{\newpage\clearpage
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$\displaystyle \; \log p(X_3|\ensuremath\boldsymbol{\Theta}_4).
$%
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$ {\cal N}_1$%
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$ {\cal N}_2$%
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$ {\cal
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{\newpage\clearpage
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$ {\cal N}_4$%
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$ [F_1,F_2]$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3597}%
$ N\times2$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3601}%
$ P(q_k)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3603}%
$ q_k$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3605}%
$ k \in
\{$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3606}%
$ ,$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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$ \}$%
\lthtmlinlinemathZ
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\stepcounter{subsubsection}
\stepcounter{subsection}
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\lthtmlinlinemathA{tex2html_wrap_indisplay3624}%
$\displaystyle X \in q_k$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3625}%
$\displaystyle \quad P(q_k|X,\ensuremath\boldsymbol{\Theta}) \geq P(q_j|X,\ensuremath\boldsymbol{\Theta}),
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$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3635}%
$ P(q_k|X,\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3637}%
$ P(q_k|\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3645}%
$\displaystyle P(q_k|X,\ensuremath\boldsymbol{\Theta}) = \frac{p(X|q_k,\ensuremath\boldsymbol{\Theta})\; P(q_k|\ensuremath\boldsymbol{\Theta})}{p(X|\ensuremath\boldsymbol{\Theta})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3653}%
$ p(X|\ensuremath\boldsymbol{\Theta})=$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3659}%
$ p(X|q_k,\ensuremath\boldsymbol{\Theta}) P(q_k|\ensuremath\boldsymbol{\Theta})$%
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\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3661}%
$\displaystyle P(q_k|X,\ensuremath\boldsymbol{\Theta}) \propto p(X|q_k,\ensuremath\boldsymbol{\Theta})\; P(q_k|\ensuremath\boldsymbol{\Theta}), \quad \forall k
$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3663}%
$ \log$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3665}%
$\displaystyle \log P(q_k|X,\ensuremath\boldsymbol{\Theta}) \propto \log p(X|q_k,\ensuremath\boldsymbol{\Theta}) + \log P(q_k|\ensuremath\boldsymbol{\Theta})$%
\lthtmlindisplaymathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3669}%
$ \ensuremath\boldsymbol{\mu}_k$%
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{\newpage\clearpage
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$ \ensuremath\boldsymbol{\Sigma}_k$%
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{\newpage\clearpage
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$ ,/u/\}$%
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{\newpage\clearpage
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$ p(X|q_k,\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3681}%
$ \log p(X|q_k,\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3689}%
$ (\ensuremath\boldsymbol{\mu}_k,\ensuremath\boldsymbol{\Sigma}_k)$%
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\stepcounter{subsubsection}
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\lthtmlinlinemathA{tex2html_wrap_inline3712}%
$ p(X|\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3720}%
$ \ensuremath\mathbf{x}_i=[F_1,F_2]^{\mathsf T}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
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$ f_k(\ensuremath\mathbf{x}_i)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3726}%
$ f_k(\ensuremath\mathbf{x}_i) = \log
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\lthtmlinlinemathZ
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$ f_{\text{/a/}}(\ensuremath\mathbf{x}_i)$%
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{\newpage\clearpage
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$ f_{\text{/e/}}(\ensuremath\mathbf{x}_i)$%
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{\newpage\clearpage
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$ f_{\text{/i/}}(\ensuremath\mathbf{x}_i)$%
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{\newpage\clearpage
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$ f_{\text{/o/}}(\ensuremath\mathbf{x}_i)$%
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{\newpage\clearpage
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$ f_{\text{/y/}}(\ensuremath\mathbf{x}_i)$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3746}%
$ [400,1800]^{\mathsf T}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3748}%
$ [400,1000]^{\mathsf T}$%
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{\newpage\clearpage
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$ [530,1000]^{\mathsf T}$%
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{\newpage\clearpage
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$ [600,1300]^{\mathsf T}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3754}%
$ [670,1300]^{\mathsf T}$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3756}%
$ [420,2500]^{\mathsf T}$%
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\lthtmlinlinemathA{tex2html_wrap_inline3815}%
$ \log p(\ensuremath\mathbf{x}_i|q_k,\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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$ x_1$%
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$ d-1$%
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$ f_k(\ensuremath\mathbf{x})$%
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$ k$%
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$\displaystyle \ensuremath\mathbf{x}\in q_k \quad \Leftrightarrow \quad f_k(\ensuremath\mathbf{x},\ensuremath\boldsymbol{\Theta}_k) \geq f_l(\ensuremath\mathbf{x},\ensuremath\boldsymbol{\Theta}_l),
\quad \forall l \neq k
$%
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{\newpage\clearpage
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$ P(q_k|\ensuremath\mathbf{x}_i)$%
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{\newpage\clearpage
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$\displaystyle \ensuremath\mathbf{x}\in q_k$%
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{\newpage\clearpage
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$\displaystyle \Leftrightarrow$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3858}%
$\displaystyle P(q_k|\ensuremath\mathbf{x}_i) \geq P(q_l|\ensuremath\mathbf{x}_i),\quad \forall l \neq k$%
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{\newpage\clearpage
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$\displaystyle p(\ensuremath\mathbf{x}_i|q_k)\; P(q_k) \geq p(\ensuremath\mathbf{x}_i|q_l)\; P(q_l),\quad
\forall l \neq k$%
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{\newpage\clearpage
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$\displaystyle \log p(\ensuremath\mathbf{x}_i|q_k)+\log P(q_k) \geq \log
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$ \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}$%
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$ k=1,\ldots,K$%
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$ \ensuremath\mathbf{x}_n$%
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{\newpage\clearpage
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$ n=1,\ldots,N$%
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$ k^{\text{th}}$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3929}%
$\displaystyle d_k(\ensuremath\mathbf{x}_n)$%
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{\newpage\clearpage
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$\displaystyle \, \left\| \ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k \right\|^2$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay3937}%
$\displaystyle (\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)^{\mathsf T}(\ensuremath\mathbf{x}_n-\ensuremath\boldsymbol{\mu}_k)$%
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{\newpage\clearpage
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$\displaystyle d_k(\ensuremath\mathbf{x}_n) \, < \, d_l(\ensuremath\mathbf{x}_n), \qquad \forall l \neq k
$%
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{\newpage\clearpage
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$\displaystyle J = \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} d_k(\ensuremath\mathbf{x}_n)
$%
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{\newpage\clearpage
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$ \{\ensuremath\boldsymbol{\mu}_{\text{/a/}},
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\ensuremath\boldsymbol{\mu}_{\text{/y/}}\}$%
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$ 2\times$%
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$ {\cal N}(\ensuremath\boldsymbol{\mu}_{k},\ensuremath\boldsymbol{\Sigma}_{k}),
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$ k=1\ldots K$%
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{\newpage\clearpage
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$ 1/K$%
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{\newpage\clearpage
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$ Q$%
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$ p(X,Q|\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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$ i$%
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$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} =$%
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{\newpage\clearpage
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$\displaystyle q_k^{(i)}
$%
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{\newpage\clearpage
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$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} =$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4002}%
$\displaystyle P(q_k^{(i+1)}|\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac{\mbox{number of training points
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$%
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{\newpage\clearpage
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$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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$\displaystyle \sum_{X} P(X|\ensuremath\boldsymbol{\Theta}) \;=\; \sum_{Q} \sum_{X} p(X,Q|\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4013}%
$\displaystyle \sum_{k=1}^{K} \sum_{\ensuremath\mathbf{x}_n \in q_k} \log p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta}_k),$%
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4020}%
$ \{\ensuremath\boldsymbol{\Sigma}_{\text{/a/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/i/}},
\ensuremath\boldsymbol{\Sigma}_{\text{/o/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/y/}}\}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4022}%
$ [P_{\text{/a/}},P_{\text{/e/}},P_{\text{/i/}},P_{\text{/o/}},P_{\text{/y/}}]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
\stepcounter{subsection}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4030}%
$ P(q_k) = 1/K$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4032}%
$ P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4036}%
$ q_k^{(i)}$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4039}%
$\displaystyle P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4043}%
$\displaystyle \frac{P(q_k^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)})
\cdot p(\ensuremath\mathbf{x}_n|q_k^{(i)},\ensuremath\boldsymbol{\Theta}^{(i)})}
{p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta}^{(i)})}$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4047}%
$\displaystyle \frac{P(q_k^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)}) \cdot p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\mu}_k^{(i)},\ensuremath\boldsymbol{\Sigma}_k^{(i)}) }
{\sum_j P(q_j^{(i)}|\ensuremath\boldsymbol{\Theta}^{(i)}) \cdot p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\mu}_j^{(i)},\ensuremath\boldsymbol{\Sigma}_j^{(i)}) }$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4051}%
$ [0,1]$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4057}%
$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} \ensuremath\mathbf{x}_n
P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})}
{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4059}%
$\displaystyle \ensuremath\boldsymbol{\Sigma}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})\;
(\ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k^{(i+1)})(\ensuremath\mathbf{x}_n - \ensuremath\boldsymbol{\mu}_k^{(i+1)})^{\mathsf T}}
{\sum_{n=1}^{N} P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)})} $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4061}%
$\displaystyle P(q_k^{(i+1)}|\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac{1}{N} \sum_{n=1}^{N}
P(q_k^{(i)}|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}^{(i)}) $%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4064}%
$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) = \log p(X|\ensuremath\boldsymbol{\Theta})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4065}%
$\displaystyle = \log \sum_Q p(X,Q|\ensuremath\boldsymbol{\Theta})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4066}%
$\displaystyle = \log \sum_Q P(Q|X,\ensuremath\boldsymbol{\Theta})p(X|\ensuremath\boldsymbol{\Theta})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4067}%
$\displaystyle = \log \sum_{k=1}^{K} P(q_k|X,\ensuremath\boldsymbol{\Theta}) p(X|\ensuremath\boldsymbol{\Theta})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4069}%
$ \left( \log \sum_j \lambda_j y_j \geq
\sum_j \lambda_j \log y_j \mbox{ if } \sum_j \lambda_j = 1 \right)$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4070}%
$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) \ge J(\ensuremath\boldsymbol{\Theta})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4071}%
$\displaystyle = \sum_{k=1}^{K} P(q_k|X,\ensuremath\boldsymbol{\Theta}) \log p(X|\ensuremath\boldsymbol{\Theta})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4072}%
$\displaystyle = \sum_{k=1}^{K} \sum_{n=1}^{N} P(q_k|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}) \log p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta})$%
\lthtmlindisplaymathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4074}%
$ J(\ensuremath\boldsymbol{\Theta})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline4076}%
$ {\cal
L}(\ensuremath\boldsymbol{\Theta})$%
\lthtmlinlinemathZ
\lthtmlcheckvsize\clearpage}
\end{document}