Cleaning model section

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.