\section{VEM}

\begin{frame}
    \frametitle{Pourquoi VE minimise KL ?}
    \begin{align*}
        \ell_c(\bY,\bZ,\bW;\theta)                               & = \log \Prob(\bZ, \bW|\bY;\theta) + \ell(\bY;\theta)                                    \\
        \Leftrightarrow            \ell(\bY;\theta)              & = \ell_c(\bY,\bZ,\bW;\theta) - \log \Prob(\bZ, \bW|\bY;\theta)                          \\
        \Leftrightarrow            \Esp_{\Ryt}[\ell(\bY;\theta)] & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\
        \Leftrightarrow            \ell(\bY;\theta)              & = \Esp_{\Ryt}[\ell_c(\bY,\bZ,\bW;\theta)] - \Esp_{\Ryt}[\log \Prob(\bZ,\bW|\bY;\theta)] \\
    \end{align*}
    \begin{align*}
        \text{Or }\KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)}                         & = - \Esp_{\Ryt} [\log \frac{\Prob(\bZ,\bW|\bY;\theta)}{\Ryt}] \\
        = - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)] +                          & \underbrace{\Esp_{\Ryt[\log \Ryt]}}_{-\Hshannon(\Ryt)}        \\
        \Leftrightarrow \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} + \Hshannon(\Ryt) & = - \Esp_{\Ryt} [\log \Prob(\bZ,\bW|\bY;\theta)]
    \end{align*}
    D'où $\ell(\bY;\theta) - \KL{\Ryt}{\log \Prob(\bZ,\bW|\bY;\theta)} = \mathcal{J}(\tau;\theta) \qed$
\end{frame}

\section{Résultats~\cite{baldockSystemsApproachReveals2019a,baldockDailyTemporalStructure2011}}
\begin{frame}[allowframebreaks]
    \begin{figure}[ht]
        \centering
        \begin{subfigure}[t]{0.5\textwidth}
            \centering
            \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Bristol.pdf}
            \caption{Donnée}
        \end{subfigure}\hfil
        \begin{subfigure}[t]{0.5\textwidth}
            \centering
            \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Bristol.pdf}
            \caption{Réordonnée}
        \end{subfigure}
        \caption{Bristol}
    \end{figure}

    \begin{figure}[ht]
        \centering
        \begin{subfigure}[t]{0.5\textwidth}
            \centering
            \includegraphics[width=0.45\textwidth]{tikz/applications/baldock/mat-Baldock2019_Edinburgh.pdf}
            \caption{Donnée}
        \end{subfigure}\hfil
        \begin{subfigure}[t]{0.5\textwidth}
            \centering
            \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Edinburgh.pdf}
            \caption{Réordonnée}
        \end{subfigure}
        \caption{Edinburgh}
    \end{figure}

    \begin{figure}
        \begin{subfigure}[ht]{0.5\textwidth}
            \centering
            \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Leeds.pdf}
            \caption{Donnée}
        \end{subfigure}\hfil
        \begin{subfigure}[ht]{0.5\textwidth}
            \centering
            \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Leeds.pdf}
            \caption{Réordonnée}
        \end{subfigure}
        \caption{Leeds}
    \end{figure}

    \begin{figure}
        \begin{subfigure}[ht]{0.5\textwidth}
            \centering
            \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/mat-Baldock2019_Reading.pdf}
            \caption{Donnée}
        \end{subfigure}\hfil
        \begin{subfigure}[ht]{0.5\textwidth}
            \centering
            \includegraphics[width=0.5\textwidth]{tikz/applications/baldock/colbisbm-mat-Baldock2019_Reading.pdf}
            \caption{Réordonnée}
        \end{subfigure}
        \caption{Reading}
    \end{figure}
\end{frame}