diff options
Diffstat (limited to 'finale')
| -rw-r--r-- | finale/project_proposal.tex | 39 |
1 files changed, 35 insertions, 4 deletions
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} |