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Louis 2025-05-29 11:18:05 +02:00
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principal.tex
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@ -277,7 +277,7 @@
\end{frame} \end{frame}
\begin{frame}{Parameter estimation}{Solution} \begin{frame}{Parameter estimation}{Solution}
\emph{Variational EM}~\cite{daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024}. \emph{Variational EM}~\cite{daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024}.
\begin{block}{Variational approximation of $\mathbf{Z,W|Y},\theta^{(t-1)}$} \begin{block}{Variational approximation of $\mathbf{Z,W|Y}$}
$\mathcal{R}_{Y^m,\tau}(Z^m, W^m) = $\mathcal{R}_{Y^m,\tau}(Z^m, W^m) =
\mathcal{R}^1_{Y^m,\tau}(Z^m) \mathcal{R}^1_{Y^m,\tau}(Z^m)
{\color{red}\times} {\color{red}\times}
@ -286,8 +286,8 @@
\begin{align*} \begin{align*}
\ell (\mathbf{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg( \ell (\mathbf{Y};\theta) \geq \color{red}\sum_{m=1}^{M} \bigg(
\color{black} \mathbb{E}_{\mathcal{R}_{Y^m,\tau}(Z^m,W^m)} \color{black} \mathbb{E}_{\mathcal{R}_{Y^m,\tau}(Z^m,W^m)}
\left[ \ell_c(Y^m,Z^m,W^m ; \theta^{(t)}) \right] + \\ \left[ \ell_c(Y^m,Z^m,W^m ; \theta) \right] + \\
\mathcal{H}(\mathcal{R}_{Y^m,\theta^{(t)}} \mathcal{H}(\mathcal{R}_{Y^m,\tau}
(Z^m, W^m)) (Z^m, W^m))
\color{red}\bigg) \color{black} \color{red}\bigg) \color{black}
\eqcolon \mathcal{J}(\mathcal{R}_{\mathbf{Y},\tau};\theta) \eqcolon \mathcal{J}(\mathcal{R}_{\mathbf{Y},\tau};\theta)