fixing small bug

1 files changed, 4 insertions, 1 deletions

diff --git a/notes/maximum_likelihood_approach.tex b/notes/maximum_likelihood_approach.tex
index 18586e8..c145b0c 100644
--- a/notes/maximum_likelihood_approach.tex
+++ b/notes/maximum_likelihood_approach.tex
@@ -1,5 +1,6 @@
 \documentclass[11pt]{article}
 \usepackage{fullpage, amsmath, amssymb, amsthm}
+\usepackage{color}
 \newtheorem{theorem}{Theorem}
 \newtheorem{lemma}{Lemma}
 \newtheorem{remark}{Remark}
@@ -109,6 +110,8 @@ We are going to explicitate a constant $\gamma$ such that: $\forall \Delta \in {
 Suppose that $\forall k$, $\mathbb{E}_{S(k)} \left[ \frac{1}{p_i} \right] \geq 1 + \alpha$, then with probability greater than $1 - 2 p e^{-2 \alpha^2 (1-c)^2p_{\min}^2 p_{\text{init}}^2 N}$, $$\forall k, \ Z_k > c \alpha p_{\text{init}} N$$
 \end{lemma}
 
+{\color{red} The following is not exactly correct!} Since the expectation must be taken conditioned on the probability that at least one parent is active such that $p_i$ is not 0! This does change things.
+
 \begin{proof}
 Let $S(k)$ denote the active set conditioned on the fact that node $k$ is active.
 \begin{align}
@@ -151,7 +154,7 @@ We conclude by union bound.
 \end{proof}
 
 \begin{proposition}
-Suppose that $\forall k,j$, $1 + \alpha \leq \mathbb{E}_{S(k, j)} \left[ \frac{1}{p_i} \right] \leq 1 + \beta$, then with probability greater than $1 - XXX$, condition~\ref{eq:RSC_condition} is met with $\gamma_n = \gamma n$ where $\gamma := p_{\text{init}}(\alpha - \beta p_{\text{init}})$
+Suppose that $\forall k,j$, $1 + \alpha \leq \mathbb{E}_{S(k, j)} \left[ \frac{1}{p_i} \right] \leq 1 + \beta$, then with probability greater than $1 - XXX$, condition~\ref{eq:RSC_condition} is met with $\gamma_n = \gamma n$ where $\gamma := p_{\text{init}}(\alpha - 16 \sqrt{s} \beta p_{\text{init}})$
 \end{proposition}