Robust Bayesian Experimental Designs in Normal Linear Models

Abstract

We address the problem of finding a design that minimizes the Bayes risk with respect to a fixed prior subject to being robust with respect to misspecification of the prior. Uncertainty in the prior is formulated in terms of having a family of priors instead of one single prior. Two different classes of priors are considered: $\Gamma_1$ is a family of conjugate priors, and a second family of priors $\Gamma_2$ is induced by a metric on the space of nonnegative measures. The family $\Gamma_1$ has earlier been suggested by Leamer and Polasek, while $\Gamma_2$ was considered by DeRobertis and Hartigan and Berger. The setup assumed is that of a canonical normal linear model with independent homoscedastic errors. Optimal robust designs are considered for the problem of estimating the vector of regression coefficients or a linear combination of the regression coefficients and also for testing and set estimation problems. Concrete examples are given for polynomial regression and completely randomized designs. A very surprising finding is that for $\Gamma_2$, the same design is optimal for a variety of different problems with different loss structures. In general, the results for $\Gamma_2$ are significantly more substantive. Our results are applicable to group decision making and reconciliation of opinions among experts with different priors.