From 7ca7bbef64db89503f943c734083bcec93e1dd58 Mon Sep 17 00:00:00 2001 From: Louis Date: Wed, 28 May 2025 20:32:22 +0200 Subject: [PATCH] Adding refs in appendix --- annexe.tex | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/annexe.tex b/annexe.tex index 55f0ebd..77a4966 100644 --- a/annexe.tex +++ b/annexe.tex @@ -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})) \\ \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*} - 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: \[ \text{BIC-L}(\hat{\theta}, \hat{\tau})= \Esp_{\mathcal{R}_{\mathbf{Y}, \hat{\tau}}}[\ell_c(\mathbf{Y},\mathbf{Z},\mathbf{W};\hat{\theta}^{\text{var}})] + \mathcal{H}(\mathcal{R}_{\mathbf{Y}, \hat{\tau}}) - \frac{1}{2} \text{pen}(\dots) \] + \cite{biernackiAssessingMixtureModel2000,daudinMixtureModelRandom2008,chabert-liddellLearningCommonStructures2024} \end{frame} \begin{frame}