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@ -263,7 +263,7 @@ Agenda
<h2 class="anchored" data-anchor-id="journaux">Journaux</h2>
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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="1765803852200" data-listing-date-modified-sort="1765803852000" 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>
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@ -316,10 +316,11 @@ Y_{i,j}&amp;\mid Z_i = q, W_j = r \sim \mathcal{F}(\alpha_{qr})
&amp; = \sum_{j=1}^{n_2} \biggl[\bigl(\tau_{jt} - \frac{1}{1+\sum_{s=1}^{R-1} \beta_s X_j} \bigr) X_j\biggr] = \sum_{j=1}^{n_2} \biggl[\bigl(\tau_{jt} - \rho_R^j \bigr) X_j\biggr]
\end{align*}</span></p>
<p>❓ Gradient mesure l’écart entre probas a posteriori et la proba a priori du groupe de référence ?</p>
<p><strong>Conclusion</strong>: Il manque l’exponentielle cette formulation ne somme pas à 1.</p>
</section>
<section id="modèle-sophie" class="level5">
<h5 class="anchored" data-anchor-id="modèle-sophie">Modèle Sophie</h5>
<p>Avec <span class="math inline">\rho_r^j = \frac{\exp{\beta_r X_j}}{\sum_{s=1}^{R} \exp{\beta_s X_j}} = \sigma(\pmb{\beta} \pmb{X})_{r,j}</span>, où <span class="math inline">\sigma</span> désigne le softmax.</p>
<p>Avec <span class="math inline">\rho_r^j = \frac{\exp{\beta_r X_j}}{\sum_{s=1}^{R} \exp{\beta_s X_j}} = \sigma(\pmb{\beta} \pmb{X})_{r,j}</span>, où <span class="math inline">\sigma</span> désigne le softmax. Mais il y a besoin de poser une contrainte sur l’un des <span class="math inline">(\beta_r)_{r=1,\dots,R}</span>, ici <span class="math inline">\beta_R = 0</span>.</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 - \log (\sum_{s=1}^{R} \exp{\beta_s X_j}))]
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