From 813792184f8901fcbfd0b4dda27a126653a3fc55 Mon Sep 17 00:00:00 2001
From: Louis Lacoste <louis.lacoste@hotmail.fr>
Date: Sun, 11 Aug 2024 11:39:26 +0200
Subject: [PATCH] =?UTF-8?q?rapport=20:=20ajout=20des=20conditions=20d'iden?=
 =?UTF-8?q?tifiabilit=C3=A9=20du=20iid?=
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---
 rapport/chapter3-structure-detection.tex | 21 +++++++++++++++++++--
 rapport/rapport.tex                      |  1 +
 2 files changed, 20 insertions(+), 2 deletions(-)

diff --git a/rapport/chapter3-structure-detection.tex b/rapport/chapter3-structure-detection.tex
index 0f6f8c7..f2062b3 100644
--- a/rapport/chapter3-structure-detection.tex
+++ b/rapport/chapter3-structure-detection.tex
@@ -757,7 +757,24 @@ colBiSBM models in %\ref{sec:network-clustering-of-simulated-networks}.
 
 \section{Model identifiability}
 \label{sec:model-identifiability}
-
 % Ici l'identifiabilité du modèle
+The goal here is to prove that if $\ell(\bm{X};\bm{\theta}) = \ell(\bm{X};\bm{\theta}')$ for any collection $\bm{X}$ then $\bm{\theta} = \bm{\theta}'$.
+
+Following the proof proposed by~\cite{chabert-liddellLearningCommonStructures2024a}, that adapted it to the collection case and~\cite{keribinEstimationSelectionLatent2015} that extended the result of~\cite{celisseConsistencyMaximumlikelihoodVariational2012} to the LBM Bernoulli model,
+we obtain the following proof of identifiability for the $iid$-colBiSBM:
+\begin{theorem}[Identifiability of $iid$-colBiSBM]
+    \label{thm:identifiability-iid}
+    The parameters $(\bm{\pi}, \bm{\rho}, \bm{\alpha})$ are
+    identifiable up to a label switching of the blocks if those
+    conditions are achieved:
+    \begin{itemize}
+        \item[(1.1)] $\exists m^*\in\{1,\dots,M\} : n^1_{m^*} \geq 2 Q_2 - 1~\text{and}~n^2_{m^*} \geq 2 Q_1 - 1$.
+        \item[(1.2)] $\forall 1\leq q \leq Q_1, \pi_q > 0$
+              and the coordinates of vector $\bm{\rho}
+                  {X^{m^*}}^T$ are distinct (where ${X^{m^*}}^T$ is the transpose of $X^{m^*}$).
+        \item[(1.3)] $\forall 1\leq r \leq Q_2, \rho_r > 0$
+              and the coordinates of vector $\bm{\pi}
+                  X^{m^*}$ are distinct.
+    \end{itemize}
+\end{theorem}
 
-\cite{chabert-liddellLearningCommonStructures2024a,celisseConsistencyMaximumlikelihoodVariational2012,keribinEstimationSelectionLatent2015}
\ No newline at end of file
diff --git a/rapport/rapport.tex b/rapport/rapport.tex
index 1242ccb..6f3465c 100644
--- a/rapport/rapport.tex
+++ b/rapport/rapport.tex
@@ -26,6 +26,7 @@
 	hypertexnames=true
 }
 
+\newtheorem{theorem}{Theorem}
 \usepackage{tocbibind} % Pour avoir des index pour table des matières, biblio
 \usepackage{geometry}
 \geometry{bmargin=25mm}