fedora - vault backup: 2026-06-09 18:45:34 - 3 files modified
Affected files: .obsidian/plugins/obsidian-livesync/data.json .obsidian/workspace.json Thèse/Axes/Phylogénie/SBM avec covariance latente.md
This commit is contained in:
Louis
parent
360c9ba327
commit
274c5db327
3 changed files with 157 additions and 185 deletions
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@ -53,7 +53,7 @@
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"syncInternalFilesIgnorePatterns": "\\/node_modules\\/, \\/\\.git\\/, \\/obsidian-livesync\\/",
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@ -68,14 +68,14 @@
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"isEncrypted": true
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@ -150,6 +150,7 @@
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"P2P_turnServers": "",
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"P2P_turnUsername": "",
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"P2P_turnCredential": "",
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12
.obsidian/workspace.json
vendored
12
.obsidian/workspace.json
vendored
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@ -197,9 +197,19 @@
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"icon": "quote-glyph",
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"title": "References"
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}
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},
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{
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"id": "6734703bbf1b5f03",
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"type": "leaf",
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"state": {
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"state": {},
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"icon": "git-pull-request",
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@ -31,238 +31,191 @@ avec $\Sigma$, la matrice de variance-covariance déterminée en fonction de l'a
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\usepackage{amsmath,amssymb}
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\usetikzlibrary{arrows.meta,positioning,shapes.geometric,calc}
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\usetikzlibrary{
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arrows.meta,
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positioning,
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shapes.geometric,
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calc,
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fit
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}
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\begin{document}
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\begin{tikzpicture}[
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font=\sffamily,
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>=Latex,
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node distance=1.5cm and 2cm,
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node distance=1cm and 0.25cm,
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directed/.style={-{Latex}, line width=0.8pt, draw=gray!75},
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bidirected/.style={-, line width=0.8pt, draw=red!75},
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base/.style={
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draw=gray!70,
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line width=0.9pt,
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align=center,
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inner sep=4pt
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draw=gray!70,
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line width=0.9pt,
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align=center,
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inner sep=4pt
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},
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prior/.style={base,rectangle,rounded corners=1pt,fill=blue!7},
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latent/.style={base,rectangle,rounded corners=6pt,fill=teal!8},
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known/.style={base,diamond,aspect=1.35,fill=orange!12},
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observed/.style={base,circle,fill=purple!8}
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observed/.style={base,circle,fill=purple!8},
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plate/.style={
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draw=black!60,
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rounded corners,
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inner sep=8pt
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}
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]
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%--------------------------------------------------
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% Hyperparameters rows
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% Row clustering
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%--------------------------------------------------
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\node[prior] (sigma) {$\sigma^2$};
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\node[known,right=of sigma] (Sigma) {$\Sigma$};
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\node[latent,below=of sigma] (Pi) {$P_i$};
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\node[latent,below=of Sigma] (Pip) {$P_{i'}$};
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\node[latent,below=of Pi] (Zi) {$Z_i$};
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\node[latent,below=of Pip] (Zip) {$Z_{i'}$};
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\node[latent,below=0.5cm of Pi] (Zi) {$Z_i$};
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%--------------------------------------------------
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% Hyperparameters columns
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%--------------------------------------------------
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\node[prior,right=4cm of Sigma] (rho) {$\rho$};
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\node[latent,below left=of rho] (Wj) {$W_j$};
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\node[latent,below right=of rho] (Wjp) {$W_{j'}$};
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%--------------------------------------------------
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% Intercept
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%--------------------------------------------------
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\node[prior] (alpha)
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at ($(Zi)!0.5!(Wj)+(0,-3)$)
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{$\alpha$};
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%--------------------------------------------------
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% Observations
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%--------------------------------------------------
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\node[observed]
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at ($(Zi)!0.5!(Wj)+(0,-1.4)$)
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(Yij) {$Y_{ij}$};
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\node[observed]
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at ($(Zi)!0.5!(Wjp)+(2.0,-1.4)$)
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(Yijp) {$Y_{ij'}$};
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\node[observed]
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at ($(Zip)!0.5!(Wj)+(-2.0,-1.4)$)
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(Yipj) {$Y_{i'j}$};
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\node[observed]
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at ($(Zip)!0.5!(Wjp)+(0,-1.4)$)
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(Yipjp) {$Y_{i'j'}$};
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%--------------------------------------------------
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% Row side
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%--------------------------------------------------
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\draw[directed] (sigma) -- (Pi);
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\draw[directed] (Sigma) -- (Pi);
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\draw[directed] (sigma) -- (Pip);
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\draw[directed] (Sigma) -- (Pip);
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\draw[directed] (Pi) -- (Zi);
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\draw[directed] (Pip) -- (Zip);
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%--------------------------------------------------
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% Column side
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% Column clustering
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%--------------------------------------------------
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\node[latent,right=2.5cm of Zi] (Wj) {$W_j$};
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\node[prior] (rho) at (Wj |- sigma) {$\rho$};
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\draw[directed] (rho) -- (Wj);
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\draw[directed] (rho) -- (Wjp);
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%--------------------------------------------------
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% Likelihood
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%--------------------------------------------------
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\foreach \y in {Yij,Yijp}
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\draw[directed] (Zi) -- (\y);
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\foreach \y in {Yipj,Yipjp}
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\draw[directed] (Zip) -- (\y);
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\foreach \y in {Yij,Yipj}
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\draw[directed] (Wj) -- (\y);
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\foreach \y in {Yijp,Yipjp}
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\draw[directed] (Wjp) -- (\y);
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\foreach \y in {Yij,Yijp,Yipj,Yipjp}
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\draw[directed] (alpha) -- (\y);
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% Observation
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%--------------------------------------------------
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% Correlation structure
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\node[observed] (Yij)
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at ($(Zi)!0.5!(Wj)+(0,-0.9)$)
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{$Y_{ij}$};
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\draw[directed] (Zi) -- (Yij);
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\draw[directed] (Wj) -- (Yij);
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\draw[bidirected] (Zi) -- (Wj);
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%--------------------------------------------------
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% Intercept
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%--------------------------------------------------
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\node[prior] (alpha)
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at ($(Yij)+(0,-2.5)$)
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{$\alpha$};
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\draw[directed] (alpha) -- (Yij);
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\draw[bidirected] (alpha) -- (Zi);
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\draw[bidirected] (alpha) -- (Zip);
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\draw[bidirected] (alpha) -- (Wj);
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\draw[bidirected] (alpha) -- (Wjp);
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\end{tikzpicture}
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\end{document}
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```
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```tikz
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\usepackage{tikz}
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\usepackage{amsmath}
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%--------------------------------------------------
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\usetikzlibrary{positioning,shapes.arrows, arrows.meta,shapes.geometric}
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% Plates
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\begin{document}
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%--------------------------------------------------
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\begin{tikzpicture}
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\tikzset{
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every path/.append style = {
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\node[plate,fit=(Pi)(Zi),label={[align=center]left:$i=1,\ldots,n_1$}] {};
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arrows = ->,
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> = stealth,},
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every node/.append style = {
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\node[plate,fit=(Wj),label={[align=center]right:$j=1,\ldots,n_2$}] {};
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shape = circle,
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draw = black,
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minimum size=3em
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\node[plate,
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},
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fit=(Yij),
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latent/.style = {
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label=below right:{$(i,j)$}] {};
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fill = lightgray
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},
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prior/.style = {
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fill = red},
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moral/.style = {
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dashed,
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> = {}, % remove arrow tip
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arrows = -, % ensure no arrows
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}}
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\node (y) {$Y$};
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\node[latent] (z) [above left = of y] {$Z$};
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\node[latent] (w) [above right = of y] {$W$};
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\node[latent] (P) [above = of z] {$P$};
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\node[prior] (sigma2) [above = of P] {$\sigma^2$};
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\node[prior] (rho) [above = of w] {$\rho_{1:R}$};
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\node[prior] (alpha) [below = of y] {$\boldsymbol{\alpha}$};
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\path (z) edge (y);
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\path (w) edge (y);
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\path (rho) edge (w);
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\path (alpha) edge (y);
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\path (P) edge (z);
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\path (sigma2) edge (P);
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% moral
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\path[moral] (z) edge (alpha);
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\path[moral] (w) edge (alpha);
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\path[moral] (z) edge (w);
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\end{tikzpicture}
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@ -643,14 +596,14 @@ $$
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\end{align*}
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$$
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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 $r$.
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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 $r$.
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Ce qui donne pour les $\tilde{\pi}_{i,k}$ de la posterior:
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$$
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\begin{align*}
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\tilde{\pi}_{i,k} = p(Z_{i} = k\mid Y_{i,\bullet},\alpha,W,P_{i}) & \propto p(Y_{i,\bullet}\mid Z_{i} = k, \alpha, W, P_{i})p(Z_{i}=k\mid P_{i})\\
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& \propto \pi_{i,k} \prod_{r=1}^{R} \alpha_{k,r}^{R_{ir}}(1-\alpha_{k,r})^{F_{ir}}
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& \propto \pi_{i,k} \prod_{r=1}^{R} \alpha_{k,r}^{R^W_{ir}}(1-\alpha_{k,r})^{F^W_{ir}}
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\end{align*}
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$$
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@ -864,6 +817,14 @@ p(Y_{\bullet,j}\mid W_j=r,\alpha,Z)&= \prod_{i=1}^{n_1} \alpha_{Z_i,r}^{Y_{i,j}}
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\end{align*}
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$$
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On peut poser $R_{jq}^{Z} = \sum_{i=1}^{n_{1}}\mathbb{1}_{Z_{i}=q} Y_{i,j}$ et $F_{jq}^{Z} = \sum_{i=1}^{n_{1}}\mathbb{1}_{Z_{i}=q} (1-Y_{i,j})$ et on définit les matrices $\mathbf{R}^Z$ et $\mathbf{F}^{Z}$.
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$$
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\begin{align*}
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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}
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\end{align*}
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$$
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**A MODIFIER**
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On a :
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Reference in a new issue