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Revert "vault backup: 2026-06-10 16:17:38"

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335
Thèse/Axes/Phylogénie/SBM avec covariance latente.md
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@ -1,4 +1,6 @@
# Inclusion dans la thèse
Ce modèle ci est une proposition pour remettre en cause l'hypothèse du LBM d'indépendance des $Z_{i}$ en définissant une structure de dépendance dépendantes de la phylogénie.
# Idée du modèle
@ -33,238 +35,191 @@ avec $\Sigma$, la matrice de variance-covariance déterminée en fonction de l'a
\usepackage{amsmath,amssymb}
\usetikzlibrary{arrows.meta,positioning,shapes.geometric,calc}
\usetikzlibrary{
arrows.meta,
positioning,
shapes.geometric,
calc,
fit
}
\begin{document}
\begin{tikzpicture}[
font=\sffamily,
>=Latex,
node distance=1.5cm and 2cm,
node distance=1cm and 0.25cm,
directed/.style={-{Latex}, line width=0.8pt, draw=gray!75},
bidirected/.style={-, line width=0.8pt, draw=red!75},
base/.style={
draw=gray!70,
line width=0.9pt,
align=center,
inner sep=4pt
draw=gray!70,
line width=0.9pt,
align=center,
inner sep=4pt
},
prior/.style={base,rectangle,rounded corners=1pt,fill=blue!7},
latent/.style={base,rectangle,rounded corners=6pt,fill=teal!8},
known/.style={base,diamond,aspect=1.35,fill=orange!12},
observed/.style={base,circle,fill=purple!8}
observed/.style={base,circle,fill=purple!8},
plate/.style={
draw=black!60,
rounded corners,
inner sep=8pt
}
]
%--------------------------------------------------
% Hyperparameters rows
% Row clustering
%--------------------------------------------------
\node[prior] (sigma) {$\sigma^2$};
\node[known,right=of sigma] (Sigma) {$\Sigma$};
\node[latent,below=of sigma] (Pi) {$P_i$};
\node[latent,below=of Sigma] (Pip) {$P_{i'}$};
\node[latent,below=of Pi] (Zi) {$Z_i$};
\node[latent,below=of Pip] (Zip) {$Z_{i'}$};
\node[latent,below=0.5cm of Pi] (Zi) {$Z_i$};
%--------------------------------------------------
% Hyperparameters columns
%--------------------------------------------------
\node[prior,right=4cm of Sigma] (rho) {$\rho$};
\node[latent,below left=of rho] (Wj) {$W_j$};
\node[latent,below right=of rho] (Wjp) {$W_{j'}$};
%--------------------------------------------------
% Intercept
%--------------------------------------------------
\node[prior] (alpha)
at ($(Zi)!0.5!(Wj)+(0,-3)$)
{$\alpha$};
%--------------------------------------------------
% Observations
%--------------------------------------------------
\node[observed]
at ($(Zi)!0.5!(Wj)+(0,-1.4)$)
(Yij) {$Y_{ij}$};
\node[observed]
at ($(Zi)!0.5!(Wjp)+(2.0,-1.4)$)
(Yijp) {$Y_{ij'}$};
\node[observed]
at ($(Zip)!0.5!(Wj)+(-2.0,-1.4)$)
(Yipj) {$Y_{i'j}$};
\node[observed]
at ($(Zip)!0.5!(Wjp)+(0,-1.4)$)
(Yipjp) {$Y_{i'j'}$};
%--------------------------------------------------
% Row side
%--------------------------------------------------
\draw[directed] (sigma) -- (Pi);
\draw[directed] (Sigma) -- (Pi);
\draw[directed] (sigma) -- (Pip);
\draw[directed] (Sigma) -- (Pip);
\draw[directed] (Pi) -- (Zi);
\draw[directed] (Pip) -- (Zip);
%--------------------------------------------------
% Column side
% Column clustering
%--------------------------------------------------
\node[latent,right=2.5cm of Zi] (Wj) {$W_j$};
\node[prior] (rho) at (Wj |- sigma) {$\rho$};
\draw[directed] (rho) -- (Wj);
\draw[directed] (rho) -- (Wjp);
%--------------------------------------------------
% Likelihood
%--------------------------------------------------
\foreach \y in {Yij,Yijp}
\draw[directed] (Zi) -- (\y);
\foreach \y in {Yipj,Yipjp}
\draw[directed] (Zip) -- (\y);
\foreach \y in {Yij,Yipj}
\draw[directed] (Wj) -- (\y);
\foreach \y in {Yijp,Yipjp}
\draw[directed] (Wjp) -- (\y);
\foreach \y in {Yij,Yijp,Yipj,Yipjp}
\draw[directed] (alpha) -- (\y);
% Observation
%--------------------------------------------------
% Correlation structure
\node[observed] (Yij)
at ($(Zi)!0.5!(Wj)+(0,-0.9)$)
{$Y_{ij}$};
\draw[directed] (Zi) -- (Yij);
\draw[directed] (Wj) -- (Yij);
\draw[bidirected] (Zi) -- (Wj);
%--------------------------------------------------
% Intercept
%--------------------------------------------------
\node[prior] (alpha)
at ($(Yij)+(0,-2.5)$)
{$\alpha$};
\draw[directed] (alpha) -- (Yij);
\draw[bidirected] (alpha) -- (Zi);
\draw[bidirected] (alpha) -- (Zip);
\draw[bidirected] (alpha) -- (Wj);
\draw[bidirected] (alpha) -- (Wjp);
\end{tikzpicture}
\end{document}
```
```tikz
\usepackage{tikz}
\usepackage{amsmath}
%--------------------------------------------------
\usetikzlibrary{positioning,shapes.arrows, arrows.meta,shapes.geometric}
% Plates
\begin{document}
%--------------------------------------------------
\begin{tikzpicture}
\tikzset{
every path/.append style = {
\node[plate,fit=(Pi)(Zi),label={[align=center]left:$i=1,\ldots,n_1$}] {};
arrows = ->,
> = stealth,},
every node/.append style = {
\node[plate,fit=(Wj),label={[align=center]right:$j=1,\ldots,n_2$}] {};
shape = circle,
draw = black,
minimum size=3em
\node[plate,
},
fit=(Yij),
latent/.style = {
label=below right:{$(i,j)$}] {};
fill = lightgray
},
prior/.style = {
fill = red},
moral/.style = {
dashed,
> = {}, % remove arrow tip
arrows = -, % ensure no arrows
}}
\node (y) {$Y$};
\node[latent] (z) [above left = of y] {$Z$};
\node[latent] (w) [above right = of y] {$W$};
\node[latent] (P) [above = of z] {$P$};
\node[prior] (sigma2) [above = of P] {$\sigma^2$};
\node[prior] (rho) [above = of w] {$\rho_{1:R}$};
\node[prior] (alpha) [below = of y] {$\boldsymbol{\alpha}$};
\path (z) edge (y);
\path (w) edge (y);
\path (rho) edge (w);
\path (alpha) edge (y);
\path (P) edge (z);
\path (sigma2) edge (P);
% moral
\path[moral] (z) edge (alpha);
\path[moral] (w) edge (alpha);
\path[moral] (z) edge (w);
\end{tikzpicture}
@ -645,14 +600,14 @@ $$
\end{align*}
$$
En posant $R_{ir}=\sum_{j=1}^{n_{2}}W_{jr}Y_{ij}$ et $F_{ir}=\sum_{j=1}^{n_{2}}W_{jr}(1-Y_{ij})$ on définit les matrices $\mathbf{R}$ et $\mathbf{F}$ qui comptent les succès et échecs par ligne $i$ et groupe $r$.
En posant $R_{ir}^W=\sum_{j=1}^{n_{2}}W_{jr}Y_{ij}$ et $F_{ir}^W =\sum_{j=1}^{n_{2}}W_{jr}(1-Y_{ij})$ on définit les matrices $\mathbf{R}^W$ et $\mathbf{F}^W$ qui comptent les succès et échecs par ligne $i$ et groupe $r$.
Ce qui donne pour les $\tilde{\pi}_{i,k}$ de la posterior:
$$
\begin{align*}
\tilde{\pi}_{i,k} = p(Z_{i} = k\mid Y_{i,\bullet},\alpha,W,P_{i}) & \propto p(Y_{i,\bullet}\mid Z_{i} = k, \alpha, W, P_{i})p(Z_{i}=k\mid P_{i})\\
& \propto \pi_{i,k} \prod_{r=1}^{R} \alpha_{k,r}^{R_{ir}}(1-\alpha_{k,r})^{F_{ir}}
& \propto \pi_{i,k} \prod_{r=1}^{R} \alpha_{k,r}^{R^W_{ir}}(1-\alpha_{k,r})^{F^W_{ir}}
\end{align*}
$$
@ -866,35 +821,33 @@ p(Y_{\bullet,j}\mid W_j=r,\alpha,Z)&= \prod_{i=1}^{n_1} \alpha_{Z_i,r}^{Y_{i,j}}
\end{align*}
$$
**A MODIFIER**
On a :
On peut poser $R_{jq}^{Z} = \sum_{i=1}^{n_{1}}\mathbb{1}_{Z_{i}=q} Y_{i,j}$ et $F_{jq}^{Z} = \sum_{i=1}^{n_{1}}\mathbb{1}_{Z_{i}=q} (1-Y_{i,j})$ et on définit les matrices $\mathbf{R}^Z$ et $\mathbf{F}^{Z}$.
$$
\begin{align*}
p(Z_{i}\mid P_{i}) & = \ilr^{-1}(P_{i}) = (\pi_{i,1},\dots,\pi_{i,k},\dots,\pi_{i,K}) \\
p(Y_{i,\bullet}\mid Z_{i}, \alpha, W) & = \prod_{j=1}^{n_{2}} \alpha_{Z_{i},W_{j}}^{Y_{ij}}(1- \alpha_{Z_{i},W_{j}})^{1-Y_{ij}} \\
p(Z_{i} = k \mid P_{i}) & = \pi_{i,k} \\
p(Y_{i,\bullet}\mid Z_{i} = k, \alpha, W) & = \prod_{j=1}^{n_{2}} \prod_{r=1}^{R}\alpha_{k,r}^{\mathbb{1}_{W_{j} = r} Y_{ij}}(1- \alpha_{k,r})^{\mathbb{1}_{W_{j} = r}(1-Y_{ij})} \\
& = \prod_{r=1}^{R} \alpha_{k,r}^{\sum_{j=1}^{n_{2}}W_{jr}Y_{ij}} (1-\alpha_{k,r})^{\sum_{j=1}^{n_{2}}W_{jr}(1-Y_{ij})}
\end{align*}
\begin{align*}
p(W_{j}=r\mid Y_{\bullet,j},\alpha, Z, \rho) \propto \rho_{r}\prod_{q=1}^Q\alpha_{q,r}^{R^Z_{jq}}(1-\alpha_{q,r})^{F_{jq}^Z}
\end{align*}
$$
En posant $R_{ir}=\sum_{j=1}^{n_{2}}W_{jr}Y_{ij}$ et $F_{ir}=\sum_{j=1}^{n_{2}}W_{jr}(1-Y_{ij})$ on définit les matrices $\mathbf{R}$ et $\mathbf{F}$ qui comptent les succès et échecs par ligne $i$ et groupe $r$.
À la fin
Ce qui donne pour les $\tilde{\pi}_{i,k}$ de la posterior:
$$
W_{j} \sim \Cat_{R}(\tilde{\rho}_{1:R}^j)
$$
### Implémentation
$$
\begin{align*}
\tilde{\pi}_{i,k} = p(Z_{i} = k\mid Y_{i,\bullet},\alpha,W,P_{i}) & \propto p(Y_{i,\bullet}\mid Z_{i} = k, \alpha, W, P_{i})p(Z_{i}=k\mid P_{i})\\
& \propto \pi_{i,k} \prod_{r=1}^{R} \alpha_{k,r}^{R_{ir}}(1-\alpha_{k,r})^{F_{ir}}
\log \tilde{p_{jr}} & = \log \rho_{r} + \sum_{q=1}^{Q} [R_{jq}^{Z} \log\alpha_{q,r} + F_{jq}^W\log{(1-\alpha_{q,r})}] \\
\tilde{\rho}_{jr} &= \frac{\exp(\log \tilde{p}_{jr} - m_{j})}{\sum_{l=1}^{R}\exp(\log \tilde{p_{jr}}-m_{j})},\quad m_{j} = \max_{l} \log p_{jr}
\end{align*}
$$
Et ainsi à la fin :
Ainsi :
$$
Z_{i}\mid P_{i}, Y, W, \alpha \sim \Cat_{K}(\tilde{\pi}_{i,1},\dots, \tilde{\pi}_{i,K})
\log \tilde{R} = \log(\rho) + \mathbf{R}^Z\log\alpha^{\top} + \mathbf{F}^Z \log(1-\alpha)^{\top}
$$
## Loi de $\alpha \mid Y,Z,W$