78 lines
No EOL
3.6 KiB
TeX
78 lines
No EOL
3.6 KiB
TeX
\section{VEM}
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\begin{frame}
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\frametitle{Why does VE minimizes KL ?}
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\begin{align*}
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\ell_c(\bY,\bZ,\bW;\theta) & = \log \Prob(\bZ, \bW|\bY;\theta) + \ell(\bY;\theta) \\
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\Leftrightarrow \ell(\bY;\theta) & = \ell_c(\bY,\bZ,\bW;\theta) - \log \Prob(\bZ, \bW|\bY;\theta) \\
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\Leftrightarrow \Esp_{\Ryt}[\ell(\bY;\theta)] & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\
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\Leftrightarrow \ell(\bY;\theta) & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\
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\end{align*}
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\begin{align*}
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\text{But }\KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} & = - \Esp_{\Ryt} [\log \frac{\Prob(\bZ,\bW|\bY;\theta)}{\Ryt}] \\
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= - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)] + & \underbrace{\Esp_{\Ryt[\log \Ryt]}}_{-\Hshannon(\Ryt)} \\
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\Leftrightarrow \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} + \Hshannon(\Ryt) & = - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)]
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\end{align*}
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Thus $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
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\end{frame}
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\section{Résultats~\cite{baldockSystemsApproachReveals2019,baldockDailyTemporalStructure2011}}
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\begin{frame}[allowframebreaks]
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol.pdf}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf}
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\caption{Reordered}
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\end{subfigure}
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\caption{Bristol}
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\end{figure}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh.pdf}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[t]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf}
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\caption{Reordered}
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\end{subfigure}
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\caption{Edinburgh}
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\end{figure}
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\begin{figure}
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds.pdf}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
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\caption{Réordonnée}
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\end{subfigure}
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\caption{Leeds}
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\end{figure}
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\begin{figure}
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading.pdf}
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\caption{Donnée}
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\end{subfigure}\hfil
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\begin{subfigure}[ht]{0.5\textwidth}
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\centering
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\includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf}
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\caption{Réordonnée}
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\end{subfigure}
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\caption{Reading}
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\end{figure}
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\end{frame} |