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Diffstat (limited to 'finale')
| -rw-r--r-- | finale/mid_report.tex | 49 | ||||
| -rw-r--r-- | finale/sparse.bib | 12 |
2 files changed, 58 insertions, 3 deletions
diff --git a/finale/mid_report.tex b/finale/mid_report.tex index e831fcc..684d0a8 100644 --- a/finale/mid_report.tex +++ b/finale/mid_report.tex @@ -5,8 +5,9 @@ \usepackage[pagebackref=false,breaklinks=true, colorlinks=true,citecolor=blue]{hyperref} \usepackage[capitalize, noabbrev]{cleveref} -\usepackage{graphicx} +\usepackage{graphicx, subfig} \usepackage{bbm} +\usepackage{fullpage} \DeclareMathOperator*{\argmax}{arg\,max} \DeclareMathOperator{\E}{\mathbb{E}} @@ -241,10 +242,50 @@ $$P(G | \Theta) \propto \exp \left( s(G)\cdot \Theta \right)$$ We can sample from the posterior by MCMC\@. This might not be the fastest solution however. We could greatly benefit from using an alternative method: \begin{itemize} -\item EM~\cite{} -\item Variational Inference~\cite{} +\item EM\@. This approach was used in \cite{linderman2014discovering} to learn +the parameters of a Hawkes process, a closely related inference problem. +\item Variational Inference. This approach was used +in~\cite{linderman2015scalable} as an extension to the paper cited in the +previous bullet point. \end{itemize} + + +\begin{figure} +\subfloat[][50 cascades]{ +\includegraphics[scale=.4]{../simulation/plots/2015-11-05_22:52:30.pdf}} +\subfloat[][100 cascades]{ +\includegraphics[scale=.4]{../simulation/plots/2015-11-05_22:52:47.pdf}}\\ +\subfloat[][150 cascades]{ +\includegraphics[scale=.4]{../simulation/plots/2015-11-05_22:53:24.pdf}} +\subfloat[][200 cascades]{ +\includegraphics[scale=.4]{../simulation/plots/2015-11-05_22:55:39.pdf}}\\ +\subfloat[][250 cascades]{ +\includegraphics[scale=.4]{../simulation/plots/2015-11-05_22:57:26.pdf}} +\subfloat[][1000 cascades]{ +\includegraphics[scale=.4]{../simulation/plots/2015-11-05_22:58:29.pdf}} +\caption{Bayesian Inference of $\Theta$ with MCMC using a $Beta(1, 1)$ prior. +For each figure, the plot $(i, j)$ on the $i^{th}$ row and $j^{th}$ column +represent a histogram of samples taken from the posterior of the corresponding +edge $\Theta_{i, j}$. The red line indicates the true value of the edge weight. +If an edge does not exist (has weight $0$) the red line is confounded with the y +axis.} +\label{betapriorbayeslearning} +\end{figure} + +\paragraph{Experiments} + +We ran some experiments on a simple network with 4 nodes with $\binom{4}{2}=6$ +parameters to learn with the MCMC package PyMC\@. We plot in +Figure~\ref{betapriorbayeslearning} the progressive learning of $\Theta$ for +increasing numbers of observations. Of note, since the IC model does not include +self-loops, the diagonal terms are never properly learned, which is expected but +not undesirable. We notice that the existence or not of an edge is (relatively) +quickly learned, with the posterior on edges with no weight converging to $0$ +after $100$ cascades. To get a concentrated posterior around the true non-zero +edge weigth requires $1000$ cascades, which is unreasonably high considering the +small number of parameters that we are learning in this experiment. + \subsection{Active Learning} In this setup, $S$ is no longer drawn from a random distribution $p_s$ but is @@ -277,5 +318,7 @@ given $(x_t)_{t\geq1}$}$ \Comment{Update posterior on $\theta$} \end{algorithmic} \end{algorithm} +\bibliography{sparse}{} +\bibliographystyle{plain} \end{document} diff --git a/finale/sparse.bib b/finale/sparse.bib index f50a0d2..9fc56df 100644 --- a/finale/sparse.bib +++ b/finale/sparse.bib @@ -504,4 +504,16 @@ year = "2009" publisher={Springer} } +@article{linderman2015scalable, + title={Scalable Bayesian Inference for Excitatory Point Process Networks}, + author={Linderman, Scott W and Adams, Ryan P}, + journal={arXiv preprint arXiv:1507.03228}, + year={2015} +} +@article{linderman2014discovering, + title={Discovering latent network structure in point process data}, + author={Linderman, Scott W and Adams, Ryan P}, + journal={arXiv preprint arXiv:1402.0914}, + year={2014} +} |