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Distributionally Robust Optimal Designs Sagnol, Guillaume


The optimal design of experiments for nonlinear (or generalized-linear) models can be formulated as the problem of finding a design $\xi$ maximizing a criterion $\Phi(\xi,\theta)$, where $\theta$ is the unknown quantity of interest that we want to determine. Several strategies have been proposed to deal with the dependency of the optimal design on the unknown parameter $\theta$. Whenever possible, a sequential approach can be applied. Otherwise, Bayesian and Maximin approaches have been proposed. The robust maximin designs maximizes the worst-case of the criterion $\Phi(\xi,\theta)$, when $\theta$ varies in a set $\Theta$. In many cases however, such a design performs well only in a very small subset of the region $\Theta$, so a maximin design might be far away from the optimal design for the true value of the unknown parameter. On the other hand, it has been proposed to assume that a prior for $\theta$ is available, and to minimize the expected value of the criterion with respect to this prior. One objection to this approach is that when a sequential approach is not possible, we rarely have precise distributional information on the unkown parameter $\theta$. In the literature on optimization under uncertainty, the Bayesian and maximin approaches are known as "stochastic programming" and "robust optimization", respectively. A third way, somehow in between the two other paradigms, has received a lot of attention recently. The distributionally robust approach can be seen as a robust counterpart of the Bayesian approach, in which we optimize against the worst-case of all priors belonging to a family of probability distributions. In this talk, we will give equivalence theorems to characterize distributionally-robust optimal (DRO) designs. We will show that DRO-designs can be computed numerically by using semidefinite programming (SDP) or second-order cone programming (SOCP), and we will compare DRO-designs to Bayesian and maximin-optimal designs in simple cases.

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