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Adding refs in appendix

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Louis 2025-05-28 20:32:22 +02:00
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@ -106,9 +106,10 @@
\Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}}[\ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta})] & = \log p(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) \\ \Esp_{\mathbf{Z}, \mathbf{W}|\mathbf{Y}}[\ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta})] & = \log p(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) \\
\text{And thus,}~\text{ICL}(\hat{\theta}) & = \log p(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) - \frac{1}{2} \text{pen}(\dots) \text{And thus,}~\text{ICL}(\hat{\theta}) & = \log p(\mathbf{Y};\hat{\theta}) - \mathcal{H}(p(\mathbf{Z},\mathbf{W}|\mathbf{Y})) - \frac{1}{2} \text{pen}(\dots)
\end{align*} \end{align*}
Recalling that $\mathbf{Z,W|Y}$ is inaccessible, we use the \emph{variational approximation} $\mathcal{R}_{\mathbf{Y},\hat{\tau}}$ and not penalizing the entropy of the distribution we derive the BIC-Like criterion: $\mathbf{Z,W|Y}$ intractable, use the \emph{variational approximation} $\mathcal{R}_{\mathbf{Y},\hat{\tau}}$ and don't penalize the entropy we derive the BIC-Like:
\[ \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) \[ \text{BIC-L}(\hat{\theta}, \hat{\tau})= \Esp_{\mathcal{R}_{\mathbf{Y}, \hat{\tau}}}[\ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta}^{\text{var}})] + \mathcal{H}(\mathcal{R}_{\mathbf{Y}, \hat{\tau}}) - \frac{1}{2} \text{pen}(\dots)
\] \]
\cite{biernackiAssessingMixtureModel2000,daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024}
\end{frame} \end{frame}
\begin{frame} \begin{frame}