From e77261ac95f8b45d23e0b33fae67760ba7537109 Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Tue, 13 Oct 2015 18:29:34 -0400
Subject: rough draft for Finale

---
 finale/project_proposal.tex | 39 +++++++++++++++++++++++++++++++++++----
 1 file changed, 35 insertions(+), 4 deletions(-)

(limited to 'finale/project_proposal.tex')

diff --git a/finale/project_proposal.tex b/finale/project_proposal.tex
index 73126b0..55d44e6 100644
--- a/finale/project_proposal.tex
+++ b/finale/project_proposal.tex
@@ -1,7 +1,7 @@
 \documentclass[10pt]{article}
-\usepackage{fullpage, amsmath, amssymb}
+\usepackage{fullpage, amsmath, amssymb, amsthm}
 
-\title{Regression Analysis in Networks}
+\title{Regression Analysis with Network data}
 \author{Jean Pouget-Abadie, Thibaut Horel}
 \date{}
 
@@ -10,8 +10,39 @@
 
 \subsection*{The Network Inference problem}
 
-\subsection*{Prior Work}
+The network inference problem concerns itself with learning the edges and the
+edge weights of an unknown network. Each edge weight is a parameter to be
+estimated. The information at our disposal is the result of a cascade process on
+the network. Here, we will focus on the Generalized Linear Cascade (GLC) model
+introduced in~\cite{}.
 
-\subsection*{Project Proposal}
+\paragraph{Short description of the GLC model}
+
+Let $X^t$ be the indicator variable of ``contagious nodes'' at time step $t$.
+A \emph{generalized linear cascade model} is a cascade model such that for each
+susceptible node $j$ in state $s$ at time step $t$, the probability of $j$
+becoming ``contagious'' at time step $t+1$ conditioned on $X^t$ is a Bernoulli
+variable of parameter $f(\theta_j \cdot X^t)$:
+\begin{equation}
+    \label{eq:glm}
+    \mathbb{P}(X^{t+1}_j = 1|X^t)
+    = f(\theta_j \cdot X^t)
+\end{equation}
+where $f: \mathbb{R} \rightarrow [0,1]$
+
+\paragraph{Problem statement}
+Assume that $X_t \sim \mathcal{D}$, where $D$ is the GLC process defined above.
+Identify the parents and estimate the edge weights for each node $i$ in the
+network $\mathcal{N}$. This can be solved using maximum likelihood:
+$$\log \mathcal{L}_i(\theta_i\,|\,x^1,\ldots,x^n) = \frac{1}{|{\cal T}_i|}
+\sum_{t\in {\cal T}_i } x_i^{t+1}\log f(\theta_i\cdot x^{t}) + (1 -
+x_i^{t+1})\log\big(1-f(\theta_i \cdot x^t)\big)$$
+
+In particular, it is known that an approximation of the variance for $\hat
+\beta$ is given by the inverse of the information matrix, which is given by:
+$$blabla$$
+
+In the case of logistic regression. In the case of the independent cascade
+model.
 
 \end{document}
-- 
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