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Louis 2026-02-16 16:25:49 +01:00
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suivi/2026-7/2026-7.qmd
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@ -101,11 +101,16 @@ V \Gamma & \approx \log((\pmb{\pi}^i)_{i=1,\dots,n_1}) = \log(\pmb{\Pi})
#### 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.
Soit $j=1,\dots,n_2$ alors
\begin{align*}
&\sigma(XB)_{j} = \sigma(XB^{\prime})_{j}\\
&\implies \exists c \in \mathbb{R}, X_{j,\bullet}B = X_{j,\bullet}B^{\prime} + c \\
&\implies \exists c \in \mathbb{R}, \begin{pmatrix} X_{j,\bullet} \beta_1 & \dots & X_{j,\bullet} \beta_{R-1} & \vec{0}_{p+1} \end{pmatrix} = X_{j,\bullet}B^{\prime} + c
&\sigma(XB) = \sigma(XB^{\prime})\\
&\implies \exists C = \begin{pmatrix}c_1 \\ \vdots \\ c_j \\ \vdots \\ c_{n_2}\end{pmatrix} \in \mathbb{R}^{n_2}, X B = X B^{\prime} + C \pmb{1}_{R}^{\top} \\
&\implies \exists C \in \mathbb{R}^{n_2}, (X B)_{j,r} = (X B^{\prime})_{j,r} + (C \pmb{1}_{R}^{\top})_{j,r} \\
&\implies \exists C \in \mathbb{R}^{n_2}, \forall r\in\{1\dots,R\}, \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\}, 0 = 0 + c_j \implies c_j = 0 \\
&\implies C = \begin{pmatrix} 0 \\ \vdots \\ 0 \end{pmatrix} \text{and thus}, B = B^{\prime} \\
\end{align*}