Deploy site [CI SKIP]
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Woodpecker CI
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a0d27d274f
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24 changed files with 182 additions and 70 deletions
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index.html
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index.html
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@ -224,14 +224,14 @@ window.Quarto = {
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<div>
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<div class="quarto-title-meta-heading">Date de publication</div>
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<div class="quarto-title-meta-contents">
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<p class="date">15 décembre 2025</p>
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<p class="date">23 décembre 2025</p>
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</div>
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</div>
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<div>
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<div class="quarto-title-meta-heading">Modifié</div>
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<div class="quarto-title-meta-contents">
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<p class="date-modified">15 décembre 2025</p>
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<p class="date-modified">23 décembre 2025</p>
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</div>
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</div>
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@ -263,7 +263,7 @@ Agenda
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<h2 class="anchored" data-anchor-id="journaux">Journaux</h2>
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<div id="listing-journal-these" class="quarto-listing quarto-listing-container-default">
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<div class="list quarto-listing-default">
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<div class="quarto-post image-right" data-index="0" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1766102400000" data-listing-file-modified-sort="1766494258619" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="8" data-listing-word-count-sort="1452">
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<div class="thumbnail"><a href="./suivi/2025-51/2025-51.html" class="no-external">
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<div class="listing-item-img-placeholder card-img-top" > </div>
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@ -283,7 +283,7 @@ Agenda
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</div>
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<div class="delink listing-description"><a href="./suivi/2025-51/2025-51.html" class="no-external">
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Passer version article flat dans Gitlab du papier et nettoyer au minimum sur une branche clean.
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✅ <strong>C’est fait</strong> Passer version article flat dans Gitlab du papier et nettoyer au minimum sur une branche clean.
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</a></div>
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</div>
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<div class="metadata">
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@ -297,7 +297,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="1" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1765497600000" data-listing-file-modified-sort="1765819663135" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="6" data-listing-word-count-sort="1010">
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<div class="thumbnail"><a href="./suivi/2025-50/2025-50.html" class="no-external">
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<div class="listing-item-img-placeholder card-img-top" > </div>
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@ -331,7 +331,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="2" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1762128000000" data-listing-file-modified-sort="1765819663134" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="4" data-listing-word-count-sort="724">
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<div class="quarto-post image-right" data-index="2" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1762128000000" data-listing-file-modified-sort="1766494258619" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="4" data-listing-word-count-sort="724">
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<div class="thumbnail"><a href="./suivi/2025-45/2025-45.html" class="no-external">
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<div class="listing-item-img-placeholder card-img-top" > </div>
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@ -365,7 +365,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="3" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1761523200000" data-listing-file-modified-sort="1765819663134" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="4" data-listing-word-count-sort="645">
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<div class="quarto-post image-right" data-index="3" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1761523200000" data-listing-file-modified-sort="1766494258618" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="4" data-listing-word-count-sort="645">
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<div class="thumbnail"><a href="./suivi/2025-44/2025-44.html" class="no-external">
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<div class="listing-item-img-placeholder card-img-top" > </div>
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@ -399,7 +399,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="4" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1760918400000" data-listing-file-modified-sort="1765819663134" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="5" data-listing-word-count-sort="903">
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<div class="quarto-post image-right" data-index="4" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1760918400000" data-listing-file-modified-sort="1766494258618" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="5" data-listing-word-count-sort="903">
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<div class="thumbnail"><a href="./suivi/2025-43/2025-43.html" class="no-external">
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<div class="listing-item-img-placeholder card-img-top" > </div>
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@ -433,7 +433,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="5" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1758240000000" data-listing-file-modified-sort="1765819663134" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="4" data-listing-word-count-sort="668">
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<div class="quarto-post image-right" data-index="5" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1758240000000" data-listing-file-modified-sort="1766494258618" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="4" data-listing-word-count-sort="668">
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<div class="thumbnail"><a href="./suivi/2025-38/2025-38.html" class="no-external">
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<div class="listing-item-img-placeholder card-img-top" > </div>
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@ -467,7 +467,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="6" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1756425600000" data-listing-file-modified-sort="1765819663134" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="4" data-listing-word-count-sort="699">
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<div class="quarto-post image-right" data-index="6" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1756425600000" data-listing-file-modified-sort="1766494258618" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="4" data-listing-word-count-sort="699">
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<div class="thumbnail"><a href="./suivi/2025-35/2025-35.html" class="no-external">
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<div class="listing-item-img-placeholder card-img-top" > </div>
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@ -501,7 +501,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="7" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1755129600000" data-listing-file-modified-sort="1765819663133" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="4" data-listing-word-count-sort="792">
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<div class="quarto-post image-right" data-index="7" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1755129600000" data-listing-file-modified-sort="1766494258617" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="4" data-listing-word-count-sort="792">
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<div class="thumbnail"><a href="./suivi/2025-33/2025-33.html" class="no-external">
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<p class="card-img-top"><img src="suivi/2025-33/figs/auc-model.png" class="thumbnail-image card-img"/></p>
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@ -536,7 +536,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="8" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1752537600000" data-listing-file-modified-sort="1765819663133" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="14" data-listing-word-count-sort="2709">
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<div class="quarto-post image-right" data-index="8" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1752537600000" data-listing-file-modified-sort="1766494258617" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="14" data-listing-word-count-sort="2709">
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<div class="thumbnail"><a href="./suivi/2025-29/2025-29.html" class="no-external">
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<div class="listing-item-img-placeholder card-img-top" > </div>
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@ -571,7 +571,7 @@ Louis Lacoste
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="9" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1751846400000" data-listing-file-modified-sort="1765819663133" data-listing-date-modified-sort="1752192000000" data-listing-reading-time-sort="4" data-listing-word-count-sort="619">
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<div class="quarto-post image-right" data-index="9" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1751846400000" data-listing-file-modified-sort="1766494258617" data-listing-date-modified-sort="1752192000000" data-listing-reading-time-sort="4" data-listing-word-count-sort="619">
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="10" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1751241600000" data-listing-file-modified-sort="1765819663133" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="3" data-listing-word-count-sort="519">
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<div class="quarto-post image-right" data-index="10" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1751241600000" data-listing-file-modified-sort="1766494258617" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="3" data-listing-word-count-sort="519">
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<div class="thumbnail"><a href="./suivi/2025-27/2025-27.html" class="no-external">
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="11" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1750377600000" data-listing-file-modified-sort="1765819663132" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="4" data-listing-word-count-sort="691">
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<div class="quarto-post image-right" data-index="11" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1750377600000" data-listing-file-modified-sort="1766494258616" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="4" data-listing-word-count-sort="691">
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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="12" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1749772800000" data-listing-file-modified-sort="1765819663132" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="6" data-listing-word-count-sort="1009">
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<div class="quarto-post image-right" data-index="12" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1749772800000" data-listing-file-modified-sort="1766494258616" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="6" data-listing-word-count-sort="1009">
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<div class="thumbnail"><a href="./suivi/2025-24/2025-24.html" class="no-external">
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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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</a>
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</div>
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</div>
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<div class="quarto-post image-right" data-index="13" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1748390400000" data-listing-file-modified-sort="1765819663132" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="4" data-listing-word-count-sort="785">
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<div class="quarto-post image-right" data-index="13" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1748390400000" data-listing-file-modified-sort="1766494258616" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="4" data-listing-word-count-sort="785">
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</div>
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</div>
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<div class="quarto-post image-right" data-index="14" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1747958400000" data-listing-file-modified-sort="1765819663131" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="5" data-listing-word-count-sort="919">
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<div class="quarto-post image-right" data-index="14" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1747958400000" data-listing-file-modified-sort="1766494258616" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="5" data-listing-word-count-sort="919">
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</div>
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<div class="quarto-post image-right" data-index="15" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1747353600000" data-listing-file-modified-sort="1765819663131" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="6" data-listing-word-count-sort="1030">
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<div class="quarto-post image-right" data-index="15" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZSUyQ0dOTg==" data-listing-date-sort="1747353600000" data-listing-file-modified-sort="1766494258615" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="6" data-listing-word-count-sort="1030">
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<div class="quarto-post image-right" data-index="16" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZQ==" data-listing-date-sort="1746748800000" data-listing-file-modified-sort="1765819663131" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="5" data-listing-word-count-sort="814">
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<div class="quarto-post image-right" data-index="16" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZQ==" data-listing-date-sort="1746748800000" data-listing-file-modified-sort="1766494258615" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="5" data-listing-word-count-sort="814">
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<div class="quarto-post image-right" data-index="17" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZQ==" data-listing-date-sort="1746144000000" data-listing-file-modified-sort="1765819663130" data-listing-date-modified-sort="1765819663000" data-listing-reading-time-sort="3" data-listing-word-count-sort="521">
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<div class="quarto-post image-right" data-index="17" data-categories="Y29sQmlTQk0lMkNpbmYlQzMlQTlyZW5jZQ==" data-listing-date-sort="1746144000000" data-listing-file-modified-sort="1766494258614" data-listing-date-modified-sort="1766494258000" data-listing-reading-time-sort="3" data-listing-word-count-sort="521">
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|
||||
|
||||
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|
||||
|
|
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|
|||
</a>
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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||||
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||||
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|
||||
|
|
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|
|||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
|
||||
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|
||||
|
|
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|
|||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
<div class="thumbnail"><a href="./suivi/2025-15/2025-15.html" class="no-external">
|
||||
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||||
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|
||||
|
|
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|
|||
</a>
|
||||
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|
||||
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||||
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|
||||
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||||
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||||
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|
|||
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||||
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||||
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<section id="todo-list" class="level2">
|
||||
<h2 class="anchored" data-anchor-id="todo-list">TODO List</h2>
|
||||
<ul>
|
||||
<li><p>Passer version article flat dans Gitlab du papier et nettoyer au minimum sur une branche clean.</p></li>
|
||||
<li><p>⚠️ IL Y A UNE TYPO SUR LE SIGNE DE L’ENTROPIE POUR LE PAPIER: <span class="math inline">- \mathcal{H}</span> au lieu de <span class="math inline">+\mathcal{H}</span></p></li>
|
||||
<li><p>✅ <strong>C’est fait</strong> Passer version article flat dans Gitlab du papier et nettoyer au minimum sur une branche clean.</p></li>
|
||||
<li><p>✅ Corrigée !⚠️ IL Y A UNE TYPO SUR LE SIGNE DE L’ENTROPIE POUR LE PAPIER: <span class="math inline">- \mathcal{H}</span> au lieu de <span class="math inline">+\mathcal{H}</span></p></li>
|
||||
<li><p>Faire tourner clustering sur Trojelsgaard</p></li>
|
||||
<li><p>Petites opérations sur les OTUs (regarder la matrice dans les yeux):</p>
|
||||
<ul>
|
||||
<li>Ranger les OTUs par variances (i.e. <code>sd(OTU_j)</code>)</li>
|
||||
<li>Dessiner les graphiques : <span class="math inline">\mathbb{V}[OTU] = f(\mathbb{E}[OTU]), \frac{\mathbb{V}[OTU]}{\mathbb{E}[OTU]^2} = f(\mathbb{E}[OTU])</span> et <span class="math inline">\frac{\mathbb{V}[OTU]}{\mathbb{E}[OTU]} = f(\mathbb{E}[OTU]) (\approx 1)</span> si les données suivent une loi de Poisson.</li>
|
||||
<li>✅ <strong>Dans un RMD sur Human Microbiome Compendium</strong> Dessiner les graphiques : <span class="math inline">\mathbb{V}[OTU] = f(\mathbb{E}[OTU]), \frac{\mathbb{V}[OTU]}{\mathbb{E}[OTU]^2} = f(\mathbb{E}[OTU])</span> et <span class="math inline">\frac{\mathbb{V}[OTU]}{\mathbb{E}[OTU]} = f(\mathbb{E}[OTU]) (\approx 1)</span> si les données suivent une loi de Poisson.
|
||||
<ul>
|
||||
<li>HMC sur-dispersés (au-dessus bissectrice)</li>
|
||||
<li>Enterotype phyloseq sous-disp</li>
|
||||
</ul></li>
|
||||
<li>Regarder la proportion de 1. taxon rares, 2. zeros.</li>
|
||||
<li>Faire des coupures selon niveaux taxonomiques et regarder si <span class="math inline">\mathbb{V}_{\text{intra}} \approx \mathbb{V}_{\text{inter}}</span></li>
|
||||
<li><em>Bonus</em>: faire ça dans qmd et voir si forge permet gitlab pages</li>
|
||||
</ul></li>
|
||||
<li><p>Faire tourner un LBM sur Human Gut et voir si ça plante sinon:</p>
|
||||
<li><p>✅ Faire tourner un LBM sur Human Gut et voir si ça plante sinon, <strong>ça plante, la ram est surchargée.</strong></p>
|
||||
<ul>
|
||||
<li>⌛ Je tente avec SparseBM de JBL sur Python</li>
|
||||
<li>Faire LBM sur niveau taxonomique grossier, initialiser avec le résultat pour un niveau plus fin et ainsi de suite.</li>
|
||||
</ul></li>
|
||||
<li><p>⌛ Prendre jeu de données exemple de phyloseq :</p>
|
||||
<ul>
|
||||
<li>✅ 😞 enterotype tourne mais pas bon résultats (semble deux blocs échantillons mais pas vu par le modèle).</li>
|
||||
<li>🕑 des jeux de données de Mahendra ne tourne pas (phase forward interminable).</li>
|
||||
</ul></li>
|
||||
<li><p>Relire <span class="citation" data-cites="peixotoHierarchicalBlockStructures2014">Peixoto (<a href="#ref-peixotoHierarchicalBlockStructures2014" role="doc-biblioref">2014</a>)</span></p>
|
||||
<ul>
|
||||
<li>Regarder les gens qui citent les travaux de Peixoto</li>
|
||||
|
|
@ -288,8 +298,12 @@ Idées
|
|||
<li>Travailler sur Fungus Tree network</li>
|
||||
<li>Comparaison covar prop avec GREMLINS multipartite sur (log(dist_phylo), fungus-tree)</li>
|
||||
<li>Trouver manière de faire un compromis : <span class="math inline">\ell(Y,Z,W;\theta) - \lambda d(C(W),C_0)</span> avec <span class="math inline">C(W)</span> le clustering seulement sur la base de la structure LBM et <span class="math inline">C_0</span> le clustering de l’arbre. Problème <span class="math inline">d</span> est une distance entre partition, comment optimiser dessus ?</li>
|
||||
<li>Mise à jour partielle des <span class="math inline">\tau</span> : ce qui pose soucis c’est les gros calculs matriciels (c’est vraiment vrai?). Donc sorte de “stochastic” VEM où on update seulement une partie des <span class="math inline">\tau</span> à chaque itération. Et échantillonnage stratifié selon l’arbre ?</li>
|
||||
<li>Chercher à formuler le problème dual (s’il existe?) de l’optimisation du LBM. Peut-être possible d’aller plus vite alors ?</li>
|
||||
<li>⌛ Mise à jour partielle des <span class="math inline">\tau</span> : ce qui pose soucis c’est les gros calculs matriciels (c’est vraiment vrai?). Donc sorte de “stochastic” VEM où on update seulement une partie des <span class="math inline">\tau</span> à chaque itération. Et échantillonnage stratifié selon l’arbre ?
|
||||
<ul>
|
||||
<li>⌛ Simulations avec <span class="math inline">n_2</span> croissant lancée sur Migale</li>
|
||||
<li>Réimplementé VE Bernoulli dans colSBM pour Bipartite et début implémentation Stochastic VE. En fait le problème des calculs matriciels <span class="math inline">Y\times(\tau^{(1)})^{\top}</span> (<span class="math inline">n_2^2</span>) donc besoin de sous-échantillonner les noeuds de l’autre dimension à mettre à jour.</li>
|
||||
</ul></li>
|
||||
<li>✅ <strong>Inutile car besoin du primal</strong> Chercher à formuler le problème dual (s’il existe?) de l’optimisation du LBM. Peut-être possible d’aller plus vite alors ? <a href="#eq-dual" class="quarto-xref">Équation 2</a></li>
|
||||
</ul>
|
||||
</div>
|
||||
</div>
|
||||
|
|
@ -340,20 +354,6 @@ Y_{i,j}&\mid Z_i = q, W_j = r \sim \mathcal{F}(\alpha_{qr})
|
|||
- \sum_{i=1}^{n_1} \tau_{iq}^{1} \log \tau_{iq}^{1} - \sum_{j=1}^{n_2} \tau_{jr}^{2} \log \tau_{jr}^{2}
|
||||
</span></p>
|
||||
<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))]
|
||||
</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}&((\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}{1+\sum_{s=1}^{R-1} \exp{\beta_s X_j}} \biggr]\\
|
||||
& = \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. 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>
|
||||
|
|
@ -366,6 +366,118 @@ Y_{i,j}&\mid Z_i = q, W_j = r \sim \mathcal{F}(\alpha_{qr})
|
|||
\end{align*}</span></p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="idée-du-problème-dual" class="level4">
|
||||
<h4 class="anchored" data-anchor-id="idée-du-problème-dual">Idée du problème dual</h4>
|
||||
<p>Les distributions variationnelles sont définies par :</p>
|
||||
<p><span class="math display">
|
||||
q(Z,W)
|
||||
=
|
||||
\prod_{i=1}^{n_1} q_i(Z_i)
|
||||
\prod_{j=1}^{n_2} q_j(W_j),
|
||||
</span></p>
|
||||
<p>avec <span class="math display">
|
||||
q_i(Z_i=q)=\tau_{iq}^{(1)},
|
||||
\qquad
|
||||
q_j(W_j=r)=\tau_{jr}^{(2)}.
|
||||
</span></p>
|
||||
<p>Les contraintes de normalisation sont : <span class="math display">
|
||||
\sum_{q=1}^Q \tau_{iq}^{(1)} = 1,
|
||||
\qquad
|
||||
\sum_{r=1}^R \tau_{jr}^{(2)} = 1.
|
||||
</span></p>
|
||||
<hr>
|
||||
<section id="lagrangien" class="level5">
|
||||
<h5 class="anchored" data-anchor-id="lagrangien">Lagrangien</h5>
|
||||
<p>Le lagrangien du problème variationnel s’écrit : <span class="math display">
|
||||
\mathcal{L}\!\left(
|
||||
\tau^{(1)},\tau^{(2)},(\lambda_i)_{i=1}^{n_1},(\mu_j)_{j=1}^{n_2}
|
||||
\right)
|
||||
=
|
||||
\mathcal{J}(\mathcal{R},\pmb{\theta})
|
||||
+
|
||||
\sum_{i=1}^{n_1} \lambda_i
|
||||
\left(1-\sum_{q=1}^Q \tau_{iq}^{(1)}\right)
|
||||
+
|
||||
\sum_{j=1}^{n_2} \mu_j
|
||||
\left(1-\sum_{r=1}^R \tau_{jr}^{(2)}\right),
|
||||
</span> où <span class="math inline">\mathcal{J}(\mathcal{R},\pmb{\theta})</span> désigne la borne inférieure variationnelle associée au modèle et aux paramètres <span class="math inline">\Theta</span>.</p>
|
||||
<hr>
|
||||
</section>
|
||||
<section id="problème-primal-conditions-doptimalité" class="level5">
|
||||
<h5 class="anchored" data-anchor-id="problème-primal-conditions-doptimalité">Problème primal (conditions d’optimalité)</h5>
|
||||
<p>En dérivant le lagrangien par rapport aux variables variationnelles <span class="math inline">\tau^{(1)}</span> et <span class="math inline">\tau^{(2)}</span>, puis en égalisant à zéro, on obtient les équations de point fixe suivantes :</p>
|
||||
<p><span class="math display">
|
||||
\tau_{iq}^{(1)}
|
||||
\propto
|
||||
\pi_q^{(t)}
|
||||
\prod_{j=1}^{n_2}
|
||||
\prod_{r=1}^{R}
|
||||
f\!\left(Y_{ij};\alpha_{qr}^{(t)}\right)^{\tau_{jr}^{(2),(t+1)}},
|
||||
\quad
|
||||
\forall i=1,\dots,n_1,\;
|
||||
q=1,\dots,Q,
|
||||
</span></p>
|
||||
<p><span class="math display">
|
||||
\tau_{jr}^{(2)}
|
||||
\propto
|
||||
\rho_r^{(t)}
|
||||
\prod_{i=1}^{n_1}
|
||||
\prod_{q=1}^{Q}
|
||||
f\!\left(Y_{ij};\alpha_{qr}^{(t)}\right)^{\tau_{iq}^{(1),(t+1)}},
|
||||
\quad
|
||||
\forall j=1,\dots,n_2,\;
|
||||
r=1,\dots,R,
|
||||
</span> où :</p>
|
||||
<ul>
|
||||
<li><span class="math inline">\pi_q^{(t)}</span> et <span class="math inline">\rho_r^{(t)}</span> sont les proportions de classes,</li>
|
||||
<li><span class="math inline">f(\cdot;\alpha_{qr})</span> est la loi d’émission du modèle,</li>
|
||||
<li><span class="math inline">\alpha_{qr}^{(t)}</span> désigne les paramètres de bloc à l’itération <span class="math inline">t</span>.</li>
|
||||
</ul>
|
||||
<hr>
|
||||
</section>
|
||||
<section id="constantes-de-normalisation" class="level5">
|
||||
<h5 class="anchored" data-anchor-id="constantes-de-normalisation">Constantes de normalisation</h5>
|
||||
<p>Les constantes de normalisation associées sont données par :</p>
|
||||
<p><span class="math display">
|
||||
T^{(1),(t)}_i
|
||||
=
|
||||
\sum_{q=1}^{Q}
|
||||
\pi_q^{(t)}
|
||||
\exp\!\left(
|
||||
\sum_{j=1}^{n_2}
|
||||
\sum_{r=1}^{R}
|
||||
\tau_{jr}^{(2)}
|
||||
\log f\!\left(Y_{ij};\alpha_{qr}^{(t)}\right)
|
||||
\right),
|
||||
</span></p>
|
||||
<p><span class="math display">
|
||||
T^{(2),(t)}_j
|
||||
=
|
||||
\sum_{r=1}^{R}
|
||||
\rho_r^{(t)}
|
||||
\exp\!\left(
|
||||
\sum_{i=1}^{n_1}
|
||||
\sum_{q=1}^{Q}
|
||||
\tau_{iq}^{(1)}
|
||||
\log f\!\left(Y_{ij};\alpha_{qr}^{(t)}\right)
|
||||
\right).
|
||||
</span></p>
|
||||
<p>Ainsi, les mises à jour normalisées s’écrivent : <span class="math display">
|
||||
\tau_{iq}^{(1)} = \frac{1}{T^{(1),(t)}_i}(\cdots),
|
||||
\qquad
|
||||
\tau_{jr}^{(2)} = \frac{1}{T^{(2),(t)}_j}(\cdots).
|
||||
</span></p>
|
||||
<hr>
|
||||
</section>
|
||||
<section id="interprétation-duale" class="level5">
|
||||
<h5 class="anchored" data-anchor-id="interprétation-duale">Interprétation duale</h5>
|
||||
<p>Les multiplicateurs de Lagrange s’identifient alors à : <span id="eq-dual"><span class="math display">
|
||||
\lambda_i = -\log T^{(1),(t)}_i - 1,
|
||||
\qquad
|
||||
\mu_j = -\log T^{(2),(t)}_j - 1,
|
||||
\tag{2}</span></span> et le problème dual consiste à minimiser une somme de fonctions de log-partition, ce qui montre que l’algorithme VEM réalise implicitement une descente sur le dual.</p>
|
||||
</section>
|
||||
</section>
|
||||
<section id="bibliographie-à-lire-à-faire" class="level4">
|
||||
<h4 class="anchored" data-anchor-id="bibliographie-à-lire-à-faire">Bibliographie: à lire, à faire</h4>
|
||||
<ul>
|
||||
|
|
|
|||
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Add table
Reference in a new issue