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Adding BIC

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Louis 2025-05-26 17:17:57 +02:00
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commit c22661ee61
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14
annexe.tex
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@ -78,10 +78,18 @@
\begin{frame}
\frametitle{On the BIC-L}
Raconter l'histoire dans l'ordre suivant :
\begin{itemize}
\item ICL = Méthode BIC (approx Laplace) sur la log complète, fait apparaître la
pénalité de complexité et pénalise l'entropie
\item ICLv = ICL mais avec les paramètres variationnels et l'entropie variationnelle
\item BIC-L = ICLv mais sans la pénalité sur l'entropie et la rajoutant à la fin
\end{itemize}
\begin{align*}
& \text{ICL}(\hat{\theta}) = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y}|\mathbf{Z},\mathbf{W};\hat{\theta})] - \frac{1}{2} \text{pen}(\dots) \\
& \text{BIC}(\hat{\theta}) = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y}|\mathbf{Z},\mathbf{W};\hat{\theta})] + \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) - \frac{1}{2} \text{pen}(\dots) \\
& \text{BIC-L}(\hat{\theta}, \hat{\tau}) = \Esp_{\mathcal{R}_{\mathbf{Y}, \hat{\tau}}}[\log \ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta}^{\text{var}})] + \mathcal{H}(\mathcal{R}_{\mathbf{Y}, \hat{\tau}}) - \frac{1}{2} \text{pen}(\dots) \\
\text{BIC}(\hat{\theta}) & = \log p(\mathbf{Y};\hat{\theta}) - \frac{1}{2} \text{pen}(\dots) \\
& = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\log p(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta})] + \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) - \frac{1}{2} \text{pen}(\dots) \\
\text{ICL}(\hat{\theta}) & = \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}} [\ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta})] - \frac{1}{2} \text{pen}(\dots) \\
\text{BIC-L}(\hat{\theta}, \hat{\tau}) & = \Esp_{\mathcal{R}_{\mathbf{Y}, \hat{\tau}}}[\ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta}^{\text{var}})] + \mathcal{H}(\mathcal{R}_{\mathbf{Y}, \hat{\tau}}) - \frac{1}{2} \text{pen}(\dots) \\
\end{align*}
\end{frame}