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Use of full rank for X in identifiability

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Louis 2026-02-17 14:39:00 +01:00
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commit e1e8c2cbdc
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suivi/2026-7/2026-7.qmd
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@ -100,7 +100,7 @@ V \Gamma & \approx \log((\pmb{\pi}^i)_{i=1,\dots,n_1}) = \log(\pmb{\Pi})
\end{align*} \end{align*}
#### Note sur l'identifiabilité (à partir JBL et réunion JA, PB, SD) #### Note sur l'identifiabilité (à partir JBL et réunion JA, PB, SD)
Soient $B,B^{\prime}$ avec $B_{\bullet,R} = B^{\prime}_{\bullet,R} = \vec{0}_{p+1}$ ~~et $X$ de rang plein tel que $X^{\top}X$ soit inversible~~ pas nécessaire. Soient $B,B^{\prime}$ avec $B_{\bullet,R} = B^{\prime}_{\bullet,R} = \vec{0}_{p+1}$ et $X$ de rang plein tel que $X^{\top}X$ soit inversible.
\begin{align*} \begin{align*}
&\sigma(XB) = \sigma(XB^{\prime})\\ &\sigma(XB) = \sigma(XB^{\prime})\\
@ -110,8 +110,8 @@ Soient $B,B^{\prime}$ avec $B_{\bullet,R} = B^{\prime}_{\bullet,R} = \vec{0}_{p+
&\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, \sum_{k=1}^{p+1} x_{j,k} \beta_{k,R} = \sum_{k=1}^{p+1} x_{j,k} \beta^{\prime}_{k,R} + c_j \\ &\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, \sum_{k=1}^{p+1} x_{j,k} \beta_{k,R} = \sum_{k=1}^{p+1} x_{j,k} \beta^{\prime}_{k,R} + c_j \\
&\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, \sum_{k=1}^{p+1} x_{j,k} \times 0 = \sum_{k=1}^{p+1} x_{j,k} \times 0 + c_j \\ &\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, \sum_{k=1}^{p+1} x_{j,k} \times 0 = \sum_{k=1}^{p+1} x_{j,k} \times 0 + c_j \\
&\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, 0 = 0 + c_j \implies c_j = 0 \\ &\implies \exists C \in \mathbb{R}^{n_2}, \forall j\in\{1,\dots,n_2\}, 0 = 0 + c_j \implies c_j = 0 \\
&\implies C = \begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix} \text{and thus}, B = B^{\prime} \\ &\implies C = \begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix} \text{and thus}, XB = XB^{\prime} \\
& \implies (X^{\top} X)^{-1}X^{\top} X B = (X^{\top} X)^{-1}X^{\top} X B^{\prime} \implies B=B^{\prime}
\end{align*} \end{align*}
#### Description du modèle hiérarchique #### Description du modèle hiérarchique