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@ -263,7 +263,7 @@ Agenda
<h2 class="anchored" data-anchor-id="journaux">Journaux</h2>
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@ -399,7 +399,7 @@ Louis Lacoste
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@ -607,7 +607,7 @@ Louis Lacoste
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<div class="quarto-post image-right" data-index="11" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1749772800000" data-listing-file-modified-sort="1765801884651" data-listing-date-modified-sort="1765801884000" data-listing-reading-time-sort="6" data-listing-word-count-sort="1009">
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<p class="card-img-top"><img src="suivi/2025-24/figs/ari-clustering-desc&asc9.png" alt="9 réseaux - ARI pour le clustering avec modèles iid, procédure descendante et descendante&ascendante" class="thumbnail-image card-img"/></p>
@ -676,7 +676,7 @@ Louis Lacoste
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@ -306,6 +306,7 @@ Y_{i,j}&amp;\mid Z_i = q, W_j = r \sim \mathcal{F}(\alpha_{qr})
<p>Plusieurs possibilités pour la définition de <span class="math inline">\rho_r^j</span></p>
<section id="modèle-1-tabouy" class="level5">
<h5 class="anchored" data-anchor-id="modèle-1-tabouy">Modèle 1 (Tabouy)</h5>
<p>Dénominateur pas correct, ne somme pas à 1.</p>
<p><span class="math inline">\rho_r^j = \frac{\exp{\beta_r X_j\mathbf{1}_{\{r\neq R\}}}}{1+\sum_{s=1}^{R-1} \beta_s X_j}, \beta_R = 0</span> et <span class="math inline">\rho_R^{j} = \frac{1}{1+\sum_{s=1}^{R-1} \beta_s X_j}</span> (pas de compréhension intuitive)</p>
<p>La partie pertinente de l’ELBO devient: <span class="math display">
P((\beta_r)_{r=1,\dots,R}, (X_j)_{j=1,\dots,n_2}, (\tau_{jr})_{\substack{j=1,\dots,n_2\\r=1,\dots,R}} ) = \sum_{j=1}^{n_2} \sum_{r=1}^{R} [\tau_{jr} (\beta_r X_j \mathbb{1}_{r\neq R} - \log (1+\sum_{s=1}^{R-1} \beta_s X_j))]
@ -324,7 +325,7 @@ Y_{i,j}&amp;\mid Z_i = q, W_j = r \sim \mathcal{F}(\alpha_{qr})
</span></p>
<p>Et on obtient la dérivée partielle par rapport à <span class="math inline">\beta_t</span> comme: <span class="math display">\begin{align*}
\dfrac{\partial P}{\partial \beta_t}&amp;((\beta_r)_{r=1,\dots,R}, (X_j)_{j=1,\dots,n_2}, (\tau_{jr})_{\substack{j=1,\dots,n_2\\r=1,\dots,R}} ) = \sum_{j=1}^{n_2} \biggl[ \tau_{jt} X_j - \frac{X_j \exp{\beta_t X_j}}{\sum_{s=1}^{R} \exp{\beta_s X_j}} \biggr]\\
&amp; = \sum_{j=1}^{n_2} \biggl[\bigl(\tau_{jt} - \sigma(\pmb{\beta} \pmb{X})_{t,j}\bigr) X_j\biggr] = \sum_{j=1}^{n_2} \biggl[\bigl(\tau_{jt} - \rho_r^j \bigr) X_j\biggr]
&amp; = \sum_{j=1}^{n_2} \biggl[\bigl(\tau_{jt} - \sigma(\pmb{\beta} \pmb{X})_{t,j}\bigr) X_j\biggr] = \sum_{j=1}^{n_2} \biggl[\bigl(\tau_{jt} - \rho_t^j \bigr) X_j\biggr]
\end{align*}</span></p>
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