From e7e5a513c38a54272e51eb97f654a5e74aaf2c8d Mon Sep 17 00:00:00 2001
From: jeanpouget-abadie <jean.pougetabadie@gmail.com>
Date: Mon, 18 May 2015 19:58:36 +0200
Subject: changed figure and slight reformulation in model

---
 paper/sections/model.tex | 26 +++++++++++---------------
 1 file changed, 11 insertions(+), 15 deletions(-)

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

diff --git a/paper/sections/model.tex b/paper/sections/model.tex
index 575f88c..fd25c27 100644
--- a/paper/sections/model.tex
+++ b/paper/sections/model.tex
@@ -289,12 +289,12 @@ measurements and not the number of cascades.
 
 \paragraph{Regularity assumptions}
 
-We would like program~\eqref{eq:pre-mle} to be convex in order to
-solve it efficiently. A sufficient condition is to
-assume that $\mathcal{L}_i$ is a concave function, which is the case if $f$ and
-$(1-f)$ are both log-concave functions. Remember that a twice-differentiable
-function $f$ is log-concave iff. $f''f \leq f'^2$. It is easy to verify this
-property for $f$ and $(1-f)$ in the Independent Cascade Model and Voter Model.
+To solve program~\eqref{eq:pre-mle} efficiently, we would like it to be convex.
+A sufficient condition is to assume that $\mathcal{L}_i$ is concave, which is
+the case if $f$ and $(1-f)$ are both log-concave. Remember that a
+twice-differentiable function $f$ is log-concave iff. $f''f \leq f'^2$.  It is
+easy to verify this property for $f$ and $(1-f)$ in the Independent Cascade
+Model and Voter Model.
 
 Furthermore, the data-dependent bounds in Section~\ref{sec:main_theorem} will
 require the following regularity assumption on the inverse link function $f$:
@@ -329,14 +329,10 @@ $\alpha\in(0,1)$.
 
 \paragraph{Convex constraints} The voter model is only defined when
 $\Theta_{i,j}\in (0,1)$ for all $(i,j)\in E$. Similarly the independent cascade
-model is only defined when $\Theta_{i,j}> 0$. One could wonder whether or not
-these constraints need to explicitly appear in the optimization
-program~\eqref{eq:pre-mle}, otherwise the program could return an estimate
-$\hat{\theta}_i$ for which the models are undefined. We claim that adding these
-constraints is unnecessary since the likelihood function $\mathcal{L}_i$ is
-equal to $-\infty$ when the parameters are outside of the domain of definition
-of the models. Hence those ``bad'' estimates will never be returned by the
-optimization program.
+model is only defined when $\Theta_{i,j}> 0$. Because the likelihood function
+$\mathcal{L}_i$ is equal to $-\infty$ when the parameters are outside of the
+domain of definition of the models, these contraints do not need to appear
+explicitly in the optimization program.
 
 In the specific case of the voter model the constraint $\sum_j \Theta_{i,j}
 = 1$ will not necessarily be verified by the estimator obtained in
@@ -345,7 +341,7 @@ constraint to be verified, in which case the results in
 Section~\ref{sec:results} still give a bound on the recovery error. If this
 constraint needs to be satisfied, then by Lagrangian duality, there exists
 a $\lambda\in \reals$ such that adding $\lambda\big(\sum_{j}\theta_j
-- 1\big)$ to the objective function of \eqref{eq:pre-mle} enforces the
+- 1\big)$ to the objective function of~\eqref{eq:pre-mle} enforces the
 constraint. Then, it suffices to apply the results of Section~\ref{sec:results}
 to the augmented objective to obtain the same recovery guarantees. Note that
 the added term is linear and will easily satisfy all the required regularity
-- 
cgit v1.2.3-70-g09d2

