From 0b13f28fa5d0ae2731c12cba5f432a3e5232c47e Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Wed, 11 Mar 2015 11:57:27 -0400
Subject: fixed typos

---
 presentation/stats/beamer_2.tex | 37 +++++++++++++++++++------------------
 1 file changed, 19 insertions(+), 18 deletions(-)

(limited to 'presentation/stats/beamer_2.tex')

diff --git a/presentation/stats/beamer_2.tex b/presentation/stats/beamer_2.tex
index ecc66ed..258b099 100644
--- a/presentation/stats/beamer_2.tex
+++ b/presentation/stats/beamer_2.tex
@@ -62,17 +62,17 @@ What do we know? What do we want to know?
 
 \begin{itemize}
 \item At $t=0$, nodes are in three possible states: susceptible, {\color{blue} infected}, {\color{red} dead}
-\pause
+
 \item At time step $t$, each {\color{blue} infected} node $i$ has a ``one-shot'' probability $p_{i,j}$ of infecting each of his susceptible neighbors $j$ at $t+1$.
-\pause
+
 \item A node stays {\color{blue}  infected} for one round, then it {\color{red} dies}
-\pause
-\item At $t=0$, each node is {\color{blue} infected} with probability $p_{\text{init}}$
-\pause
+
+\item At $t=0$, each node is {\color{blue} infected} with probability $p_{\text{init}}$ and susceptible with probability $1-p_{\text{init}}$
+
 \item Process continues until random time $T$ when no more nodes can become infected. 
-\pause
+
 \item $X_t$: set of {\color{blue}  infected} nodes at time $t$
-\pause
+
 \item A {\bf cascade} is an instance of the ICC model: $(X_t)_{t=0, t=T}$
 \end{itemize}
 
@@ -112,7 +112,7 @@ What do we know? What do we want to know?
 
 \begin{itemize}
 \item If the {\color{orange} orange} node and the {\color{green} green} node are infected at $t=0$, what is the probability that the {\color{blue} blue} node is infected at $t=1$?
-$$1 - \mathbb{P}(\text{not infected}) = 1 - (1 - .45)(1-.04)$$
+$$1 - \mathbb{P}(\text{not infected}) = 1 - (1 - .72)(1-.04)$$
 \end{itemize}
 
 
@@ -139,7 +139,7 @@ $$\mathbb{P}(j \text{ becomes infected at t+1}|X_{t}) = 1 - \prod_{i \in {\cal N
 
 \begin{frame}
 \frametitle{Independent Cascade Model}
-For each susceptible node $j$, the event that it becomes {\color{blue} infected} conditioned on previous time step is a Bernoulli:
+For each susceptible node $j$, the event that it becomes {\color{blue} infected} conditioned on previous time step is a Bernoulli variable:
 $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$
 \begin{itemize}
 \item $\theta_{i,j} := \log(1 - p_{i,j})$
@@ -159,7 +159,7 @@ $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$
 
 \begin{frame}
 \frametitle{Independent Cascade Model}
-For each susceptible node $j$, the event that it becomes {\color{blue} infected} conditioned on previous time step is a Bernoulli:
+For each susceptible node $j$, the event that it becomes {\color{blue} infected} conditioned on previous time step is a Bernoulli variable:
 $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$
 
 \begin{block}{Decomposability}
@@ -197,7 +197,8 @@ $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$
 \begin{frame}
 \frametitle{Sparse Recovery}
 \begin{figure}
-\includegraphics[scale=.6]{../images/sparse_recovery_illustration_copy.pdf}
+\centering
+\includegraphics[scale=.6]{../images/drawing.pdf}
 \caption{$\mathbb{P}(j \in X_{t+1}| X_t) = f(X_t\cdot \theta)$}
 \end{figure}
 \end{frame}
@@ -221,7 +222,7 @@ $$(j \in X_{t+1} | X_t) \sim {\cal B} \big(f(X_t \cdot \theta_j) \big)$$
 \end{align*}
 \end{block}
 
-\begin{block}{MLE}
+\begin{block}{Penalized log-likelihood}
 For each node $i$, $$\theta_i \in \arg \max \frac{1}{n_i}{\cal {L}}_i(\theta_i | X_1, X_2, \dots, X_{n_i}) - \lambda \|\theta_i\|_1$$
 \end{block}
 
@@ -242,7 +243,7 @@ For each node $i$, $$\theta_i \in \arg \max \frac{1}{n_i}{\cal {L}}_i(\theta_i |
 \begin{itemize}
 \item Want ${\cal H}$, the hessian of ${\cal L}$ with respect to $\theta$, to be well-conditioned.
 \item $ n < dim(\theta) \implies {\cal H}$ is degenerate.
-\item {\bf Restricted Eigenvalue condition} = ``almost invertible'' on sparse vectors. 
+\item {\bf Restricted Eigenvalue condition} = invertible on ``almost sparse'' vectors. 
 \end{itemize}
 \end{block}
 
@@ -257,7 +258,7 @@ For each node $i$, $$\theta_i \in \arg \max \frac{1}{n_i}{\cal {L}}_i(\theta_i |
 For a set $S$,
 $${\cal C} := \{ \Delta : \|\Delta\|_2 = 1, \|\Delta_{\bar S}\|_1 \leq 3 \| \Delta_S\|_1 \}$$
 ${\cal H}$ verifies the $(S, \gamma)$-RE condition if:
-$$\forall \Delta \in {\cal C}, \Delta {\cal H} \Delta \geq \gamma$$
+$$\forall \Delta \in {\cal C}, \Delta^T {\cal H} \Delta \geq \gamma$$
 \end{definition}
 \end{frame}
 %%%%%%%%%%%%%%%%%%%%%%%%
@@ -310,7 +311,7 @@ By thresholding $\hat \theta_i$, if $n > C' s \log m$, we recover the support of
 
 \begin{block}{From Hessian to Gram Matrix}
 \begin{itemize}
-\item If $\log f$ and $\log 1 -f$ are strictly log-concave with constant $c$, then if $(S, \gamma)$-RE holds for the gram matrix $\frac{1}{n}X X^T$ , then $(S, c \gamma)$-RE holds for ${\cal H}$
+\item If $f$ and $1 -f$ are strictly log-concave with constant $c$, then if $(S, \gamma)$-RE holds for the gram matrix $\frac{1}{n}X X^T$ , then $(S, c \gamma)$-RE holds for ${\cal H}$
 \end{itemize}
 \end{block}
 
@@ -362,11 +363,11 @@ requires ${\cal O}(s \log (n/s)/\log C)$ measurement.
 \begin{frame}
 \frametitle{Voter Model}
 \begin{itemize}
-\pause
+
 \item {\color{red} Red} and {\color{blue} Blue} nodes. At every step, each node $i$ chooses one of its neighbors $j$ with probability $p_{j,i}$ and adopts that color at $t+1$
-\pause
+
 \item If {\color{blue} Blue} is `contagious' state:
-\pause
+
 \begin{equation}
 \nonumber
 \mathbb{P}(i \in X^{t+1}|X^t) = \sum_{j \in {\cal N}(i)\cap X^t} p_{ji} = X^t \cdot \theta_i
-- 
cgit v1.2.3-70-g09d2

