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Diffstat (limited to 'paper/sections/model.tex')
| -rw-r--r-- | paper/sections/model.tex | 8 |
1 files changed, 2 insertions, 6 deletions
diff --git a/paper/sections/model.tex b/paper/sections/model.tex index fd25c27..532ee5e 100644 --- a/paper/sections/model.tex +++ b/paper/sections/model.tex @@ -140,13 +140,9 @@ the recovery error on $\Theta_j$ is an upper bound on the error on the original $p_j$ parameters. \begin{lemma} + \label{lem:transform} $\|\hat{\theta} - \theta^* \|_2 \geq \|\hat{p} - p^*\|_2$. \end{lemma} -\begin{proof} -Using the inequality $\forall x>0, \; \log x \geq 1 - \frac{1}{x}$, we have -$|\log (\frac{1}{1 - p}) - \log (\frac{1}{1-p'})| \geq \max(1 - \frac{1-p}{1-p'}, -1 - \frac{1-p'}{1-p}) \geq \max( p-p', p'-p)$. -\end{proof} \subsubsection{The Linear Voter Model} @@ -334,7 +330,7 @@ $\mathcal{L}_i$ 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} +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 \eqref{eq:pre-mle}. In some applications, the experimenter might not need this constraint to be verified, in which case the results in |