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\usepackage{graphicx}
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\title{Bayesian and Active Learning for Graph Inference} % Poster title
\author{Thibaut Horel, Jean Pouget-Abadie} % Author(s)

\begin{document}
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\begin{frame}[t]
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\begin{block}{Problem}
  \emph{How to recover an unknown network from the observation of contagion
  cascades?}
  \vspace{1em}
  \begin{itemize}
      \item \textbf{Observe:} state (infected or not) of nodes over time.
      \item \textbf{Objective:} learn $\Theta$, matrix of edge weights.
\end{itemize}
\end{block}
\vspace{1cm}

\begin{block}{\bf Contagion Model~\cite{}}
\begin{itemize}
    \item $X^t\in\{0,1\}^N$: state of the network at time $t$
    \item At $t=0$, $X^0$ drawn from \emph{source distribution}
    \item For $t=1,2,\dots$:
        \begin{itemize}
            \item $X^t$ only depends on $X^{t-1}$
            \item for each node $j$, new state drawn independently with:
                \begin{displaymath}
                    \mathbb{P}(X^{t+1}_j = 1 | X^t) = f(\Theta_j \cdot X^t)
                \end{displaymath}
            ($f$: link function of the cascade model)
\end{itemize}
\end{itemize}
  \vspace{1em}
\begin{figure}
  \centering
\end{figure}
\end{block}
  \includegraphics[scale=1.5]{drawing.pdf}

\begin{block}{MLE}
  \begin{itemize}
    \item Log-likelihood is concave for common contagion models (IC model): SGD
      on $\{\theta_{ij}\}$
  \end{itemize}
  \vspace{1cm}
  \begin{equation*}
    \begin{split}
      \hat{\theta}\in \arg\max_\theta \sum_{t}~& y^t\log f(\theta\cdot x^t)
      \\ & + (1-y^t) \log \big(1 - f(\theta\cdot x^t)\big)
  \end{split}
\end{equation*}
\end{block}

\begin{block}{Bayesian Framework}
\begin{figure}
  \centering
  \includegraphics[scale=3]{graphical.pdf}
\end{figure}
\begin{itemize}
  \item $\phi$: fixed source distribution
  \item capture uncertainty on each edge
  \item encode expressive graph priors (tied parameters)
\end{itemize}
\end{block}

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\begin{block}{Active Learning}
  \emph{Can we gain by choosing the source node? If so, how to best choose the
  source node?}
\end{block}

\begin{block}{Heuristic 1}
  \begin{itemize}
    \item Choose source proportional to estimated degree
    \item Intuition:
      \begin{itemize}
        \item deeper cascades $\implies$ more data
        \item easier to learn non-edges than edges. Higher degree $\implies$
          more edge realizations at the same price.
      \end{itemize}
  \end{itemize}
\end{block}

\begin{block}{Heuristic 2}
  \begin{itemize}
    \item Choose source proportional to mutual information
    \begin{equation*}
      I((X_t) ,\Theta | x^0 = i) = - H(\Theta | (X_t), X_0 = i) + H(\Theta)
    \end{equation*}
    \item Exact strategy requires knowing true distribution of $(X_t)$
    \item Use estimated $\Theta$ to compute $H(\Theta | (X_t), X_0 = i)$
\end{itemize}
\end{block}

\begin{block}{Heuristic 3}
  \begin{itemize}
    \item Choose source proportional to mutual information of first step of
      cascade and $\Theta$:
  \end{itemize}
\end{block}

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%	REFERENCES
%-----------------------------------------------------------------------------

\begin{block}{References}
  {\scriptsize
  \bibliography{../../paper/sparse}
\bibliographystyle{plain}}
\end{block}

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\end{document}