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\title{Regression Analysis with Network data}
\author{Jean Pouget-Abadie, Thibaut Horel}
\date{}

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\maketitle

\subsection*{The Network Inference problem}

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{}.

\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.

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