www.pudn.com > Gaussian.zip > images.tex, change:2004-05-18,size:37831b


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A Tutorial for the Course <em>Computational Intelligence</em>\end{rawhtml}



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% !!! IMAGES START HERE !!!

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$\displaystyle p(X|\ensuremath\boldsymbol{\Theta}) = \prod_{i=1}^{N} p(\ensuremath\mathbf{x}_i|\ensuremath\boldsymbol{\Theta}) \quad \Leftrightarrow \quad 
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$\displaystyle \frac{1}{\sqrt{2\pi}^d \sqrt{\det\left(\ensuremath\boldsymbol{\Sigma}\right)}}
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$\displaystyle \frac{1}{2} \left[-d \log \left( 2\pi \right)
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$\displaystyle p(x|\ensuremath\boldsymbol{\Theta}_1) > p(x|\ensuremath\boldsymbol{\Theta}_2) \quad \Leftrightarrow \quad 
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$ {\cal N}_1: \; \ensuremath\boldsymbol{\Theta}_1 = \left( 
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\left[\begin{array}{cc}8000 & 0 \\0 & 8000\end{array}\right] 
\right)$%
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$ {\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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\lthtmlinlinemathA{tex2html_wrap_inline3537}%
$ {\cal N}_3: \; \ensuremath\boldsymbol{\Theta}_3 = \left( 
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\left[\begin{array}{cc}8000 & 8400 \\8400 & 18500\end{array}\right] 
\right)$%
\lthtmlinlinemathZ
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_inline3539}%
$ {\cal N}_4: \; \ensuremath\boldsymbol{\Theta}_4 = \left( 
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\left[\begin{array}{cc}8000 & 8400 \\8400 & 18500\end{array}\right] 
\right)$%
\lthtmlinlinemathZ
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\lthtmlinlinemathA{tex2html_wrap_indisplay3549}%
$\displaystyle \log p(X_3|\ensuremath\boldsymbol{\Theta}_1),\; \log p(X_3|\ensuremath\boldsymbol{\Theta}_2),\; \log p(X_3|\ensuremath\boldsymbol{\Theta}_3),\;$%
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{\newpage\clearpage
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$\displaystyle \; \log p(X_3|\ensuremath\boldsymbol{\Theta}_4). 
$%
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$\displaystyle X \in q_k$%
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$\displaystyle \quad P(q_k|X,\ensuremath\boldsymbol{\Theta}) \geq P(q_j|X,\ensuremath\boldsymbol{\Theta}), 
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$\displaystyle P(q_k|X,\ensuremath\boldsymbol{\Theta}) \propto p(X|q_k,\ensuremath\boldsymbol{\Theta})\; P(q_k|\ensuremath\boldsymbol{\Theta}), \quad \forall k 
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$ p(X|q_k,\ensuremath\boldsymbol{\Theta})$%
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$ \log p(X|q_k,\ensuremath\boldsymbol{\Theta})$%
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$ (\ensuremath\boldsymbol{\mu}_k,\ensuremath\boldsymbol{\Sigma}_k)$%
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$ [420,2500]^{\mathsf T}$%
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$ \{\ensuremath\boldsymbol{\mu}_{\text{/a/}}, 
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$ \{\ensuremath\boldsymbol{\Sigma}_{\text{/a/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/e/}}, \ensuremath\boldsymbol{\Sigma}_{\text{/i/}}, 
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$\displaystyle \ensuremath\boldsymbol{\mu}_{k}^{(i+1)} = \frac{\sum_{n=1}^{N} \ensuremath\mathbf{x}_n 
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$\displaystyle P(q_k^{(i+1)}|\ensuremath\boldsymbol{\Theta}^{(i+1)}) = \frac{1}{N} \sum_{n=1}^{N} 
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$\displaystyle {\cal L}(\ensuremath\boldsymbol{\Theta}) = \log p(X|\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4047}%
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{\newpage\clearpage
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{\newpage\clearpage
\lthtmlinlinemathA{tex2html_wrap_indisplay4052}%
$\displaystyle = \sum_{k=1}^{K} \sum_{n=1}^{N} P(q_k|\ensuremath\mathbf{x}_n,\ensuremath\boldsymbol{\Theta}) \log p(\ensuremath\mathbf{x}_n|\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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$ J(\ensuremath\boldsymbol{\Theta})$%
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{\newpage\clearpage
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$ {\cal 
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\end{document}