diff --git a/Thèse/Axes/Phylogénie/SBM avec covariance latente.md b/Thèse/Axes/Phylogénie/SBM avec covariance latente.md
index 792f53b..925bb80 100644
--- a/Thèse/Axes/Phylogénie/SBM avec covariance latente.md	
+++ b/Thèse/Axes/Phylogénie/SBM avec covariance latente.md	
@@ -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
 
@@ -33,239 +35,192 @@ 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}
-
-\begin{document}
-
-\begin{tikzpicture}
-
-\tikzset{
-
-every path/.append style = {
-
-arrows = ->,
-
-> = stealth,},
-
-every node/.append style = {
-
-shape = circle,
-
-draw = black,
-
-minimum size=3em
-
-},
-
-latent/.style = {
-
-fill = lightgray
-
-},
-
-prior/.style = {
-
-fill = red},
-
-moral/.style = {
-
-dashed,
-
-> = {}, % remove arrow tip
-
-arrows = -, % ensure no arrows
-
-}}
 
   
 
-\node (y) {$Y$};
+%--------------------------------------------------
+
+% Plates
+
+%--------------------------------------------------
 
   
 
-\node[latent] (z) [above left = of y] {$Z$};
+\node[plate,fit=(Pi)(Zi),label={[align=center]left:$i=1,\ldots,n_1$}] {};
 
   
 
-\node[latent] (w) [above right = of y] {$W$};
+\node[plate,fit=(Wj),label={[align=center]right:$j=1,\ldots,n_2$}] {};
 
   
 
-\node[latent] (P) [above = of z] {$P$};
+\node[plate,
+
+fit=(Yij),
+
+label=below right:{$(i,j)$}] {};
 
   
 
-\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}
 
 \end{document}
@@ -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 $\alpha \mid Y,Z,W$