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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} |