From 149b94cdbfedf43274bc783ed32f60bfbe3842ef Mon Sep 17 00:00:00 2001
From: Thibaut Horel <thibaut.horel@gmail.com>
Date: Sun, 1 Feb 2015 18:33:52 -0500
Subject: Polishing Generalizing Linear Cascades

---
 paper/sections/model.tex | 20 ++++++++++++++++++--
 1 file changed, 18 insertions(+), 2 deletions(-)

(limited to 'paper/sections/model.tex')

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index 6acfade..5f03199 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -17,14 +17,30 @@ problems for a large class of well-known cascade models.
 
 \subsection{Generalized Linear Cascade Models}
 
-Denoting by $X^t$ the state of the cascade at time step $t-1$, we interpret $X^t_j = 1$ for some node $j\in V$ as ``node $j$ exhibits the source nodes' behavior at time step $t+1$''. By the Markov property of cascades, $$\mathbb{P}(X^t | X^0, X^1, \dots, X^{t-1}) = \mathbb{P}(X^t | X^{t-1})$$
-We draw inspiration from \emph{generalized linear models} (GLM) to define the following:
+Denoting by $X^t$ the state of the cascade at time step $t-1$, we interpret
+$X^t_j = 1$ for some node $j\in V$ as ``node $j$ exhibits the source nodes'
+behavior at time step $t+1$''.  We draw inspiration from \emph{generalized
+linear models} (GLM) to define a generalized linear cascade.
 
 \begin{definition}
+    Let us denote by $\{\mathcal{F}_t, t\in\ints\}$ the natural filtration
+    induced by $\{X_t, t\in\ints\}$. A \emph{generalized linear cascade} is
+    characterized by the following equation:
+    \begin{displaymath}
+        \P[X^{t+t}=x\,|\, \mathcal{F}_t] =
+        \prod_{i\in V} f(\inprod{\theta_i}{X^{t}})^{x_i}
+        \big(1-f(\inprod{\theta_i}{X^{t}}\big)^{1-x_i}
+    \end{displaymath}
+where $f:\mathbb{R}\to[0,1]$.
 Let $f: \mathbb{R} \leftarrow \mathbb{R}$ be an inverse link function. Let ${\cal F}_{< t}$ be the filtration defined by $\{ X^0, X^1, \dots, X^{t-1} \}$. A \emph{generalized linear cascade} is defined as:
 $$\mathbb{P}(X^t = 1|{\cal F}_{< t}) = \exp(blabla$$
 \end{definition}
 
+It follows immediately from this definition that a generalized linear cascade
+satisfies the Markov property:
+\begin{displaymath}
+    \P[X^{t+1}=x\,|\mathcal{F}_t] = \P[X^{t+1}=x\,|\, X^t]
+\end{displaymath}
 \subsection{Examples}
 \label{subsec:examples}
 
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