On the number of support points of maximin and Bayesian optimal designs

TL;DR

This paper provides analytical tools to explain why maximin and Bayesian D-optimal designs for nonlinear regression models tend to have more support points as uncertainty increases, clarifying observed empirical phenomena.

Contribution

It introduces analytic methods to explain the relationship between parameter uncertainty and the support points of optimal designs, especially comparing maximin and Bayesian criteria.

Findings

Support points increase with parameter uncertainty.

Maximin designs generally have more support points than Bayesian designs.

Analytic tools explain empirical observations in design theory.

Abstract

We consider maximin and Bayesian $D$-optimal designs for nonlinear regression models. The maximin criterion requires the specification of a region for the nonlinear parameters in the model, while the Bayesian optimality criterion assumes that a prior for these parameters is available. On interval parameter spaces, it was observed empirically by many authors that an increase of uncertainty in the prior information (i.e., a larger range for the parameter space in the maximin criterion or a larger variance of the prior in the Bayesian criterion) yields a larger number of support points of the corresponding optimal designs. In this paper, we present analytic tools which are used to prove this phenomenon in concrete situations. The proposed methodology can be used to explain many empirically observed results in the literature. Moreover, it explains why maximin $D$-optimal designs are usually supported at more points than Bayesian $D$-optimal designs.