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| -rw-r--r-- | finale/sections/model.tex | 31 |
1 files changed, 16 insertions, 15 deletions
diff --git a/finale/sections/model.tex b/finale/sections/model.tex index cb8b699..3fe316a 100644 --- a/finale/sections/model.tex +++ b/finale/sections/model.tex @@ -30,20 +30,20 @@ we have: where $\bt_i$ is the $i$th column of $\Theta$. The function $f:\R\to[0,1]$ can be interpreted as the inverse link function of the model. Finally, the transitions in \cref{eq:markov} occur independently for each $i$. A cascade -continues until no infected nodes remains. +continues until no infected nodes remains. As noted in \cite{pouget} many +commonly studied contagion models can be cast as specific instances of the GLC +model. -We refer the reader to \cite{pouget} for a more complete description of the -model and examples of common contagion models which can be interpreted as -specific instances of the GLC model. - -It follows from Section 2, that a source distribution $p_s$ and -\cref{eq:markov} together completely specify the distribution $p$ of a cascade -$\mathbf{x} = (x_t)_{t\geq 0}$: +\Cref{eq:markov} and a source distribution $p_s$ together completely specify +the probability distribution of a cascade $\mathbf{x} = (x_t)_{t\geq 0}$ given +$\Theta$ and allow us to write the log-likelihood of the model: \begin{equation} \label{eq:dist} - \mathcal{L}_{\Theta}(\bx) - = p_s(x^0)\prod_{\substack{t\geq 1 \\ i\in S_t}} - f(\bt_i\cdot x^{t-1})^{x^t_i}\big(1-f(\theta_i\cdot x^{t-1})\big)^{1-x_i^t} + \begin{split} + \mathcal{L}(\Theta\,|\, \mathbf{x}) = &\log p_s(x^0)\\ + & + \sum_{t\geq 1}\sum_{i\in S_t}\Big(x_i^t\log f(\bt_i\cdot x^{t-1})\\ + &+ (1-x_i^t)\log\big(1-f(\bt_i\cdot x^{t-1})\big)\Big) + \end{split} \end{equation} \paragraph{MLE estimation.} @@ -57,12 +57,13 @@ the next time step, the MLE estimator for $\bt_i$ is obtained by solving the following optimization problem: \begin{equation} \label{eq:mle} - \hat{\theta}\in \argmax_\theta \sum_{t} y^t\log f(\theta\cdot x^t) - + (1-y^t) \log \big(1 - f(\theta\cdot x^t)\big) + \begin{split} + \hat{\bt}_i\in \argmax_\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} It is interesting to note that at the node-level, doing MLE inference for the GLC model is exactly amounts to fitting a Generalized Linear Model. When $f$ is log-concave as is the case in most examples of GLC models, then the above optimization problem becomes a convex optimization problem which can be solved -exactly and efficiently. The code to perform MLE estimation can be found in the -appendix, file \textsf{mle.py}. +exactly and efficiently. |