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diff --git a/paper/sections/model.tex b/paper/sections/model.tex index f1d9585..575f88c 100644 --- a/paper/sections/model.tex +++ b/paper/sections/model.tex @@ -326,3 +326,27 @@ for some $\alpha\in(0, 1)$ and for all $z_x\defeq\inprod{\theta^*}{x}$ such that $f(z_x)\notin\{0,1\}$. It is again easy to see that this condition is verified for the Independent Cascade Model and the Voter model for the same $\alpha\in(0,1)$. + +\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. + +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 +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 +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 +assumptions. |