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Louis 2025-12-15 14:03:51 +01:00
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suivi/2025-50/2025-50.qmd
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@ -71,6 +71,8 @@ Plusieurs possibilités pour la définition de $\rho_r^j$
##### Modèle 1 (Tabouy) ##### Modèle 1 (Tabouy)
Dénominateur pas correct, ne somme pas à 1.
$\rho_r^j = \frac{\exp{\beta_r X_j\mathbf{1}_{\{r\neq R\}}}}{1+\sum_{s=1}^{R-1} \beta_s X_j}, \beta_R = 0$ et $\rho_R^{j} = \frac{1}{1+\sum_{s=1}^{R-1} \beta_s X_j}$ (pas de compréhension intuitive) $\rho_r^j = \frac{\exp{\beta_r X_j\mathbf{1}_{\{r\neq R\}}}}{1+\sum_{s=1}^{R-1} \beta_s X_j}, \beta_R = 0$ et $\rho_R^{j} = \frac{1}{1+\sum_{s=1}^{R-1} \beta_s X_j}$ (pas de compréhension intuitive)
La partie pertinente de l'ELBO devient: La partie pertinente de l'ELBO devient:
@ -98,7 +100,7 @@ $$
Et on obtient la dérivée partielle par rapport à $\beta_t$ comme: Et on obtient la dérivée partielle par rapport à $\beta_t$ comme:
\begin{align*} \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 \exp{\beta_t X_j}}{\sum_{s=1}^{R} \exp{\beta_s X_j}} \biggr]\\ \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 \exp{\beta_t X_j}}{\sum_{s=1}^{R} \exp{\beta_s X_j}} \biggr]\\
& = \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] & = \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*} \end{align*}