A new approach to optimal design under model uncertainty motivated by multi-armed bandits

An optimal design is usually model-dependent and is sub-optimal if the postulated model is not correctly specified. Furthermore, it is far from ideal even if it is efficient for model selection but has a poor performance for estimating parameters in the selected model. In practice, it is common that a researcher has a list of candidate models at hand and a design has to be found that is efficient for both model discrimination and parameter estimation in the (unknown) “true” model. In this article, we use a multi-armed bandits approach to balance these two competing goals in the design of experiments. We develop a sequential algorithm to provide a design that has asymptotically the same performance as an optimal design when the “true” model could be correctly specified in advance. A lower bound is established to quantify the relative efficiency between the proposed design and an optimal design for the “true” model. Some comparisons with other state-of-the-art algorithms for model discrimination and parameter estimation are discussed. The advantages of the proposed method are illustrated by several numerical examples.