Bayesian Designs for Maximizing Information and Outcome

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Abstract
This article deals with a novel utility function to design experiments in a Bayesian framework. This utility function is a linear combination of the gain in Shannon information and of the total outcome of the experiment, defined as the sum of observed values in the dependent variable of a linear model. Thus the expected posterior utility to be maximized is a combination of the Bayes D-optimality criterion and the posterior expectation of the total output. Earlier studies have shown that Bayesian parallels of the classical D-, A-, and E-optimal designs can be obtained by considering utility or loss functions concerned with efficient estimation of the parameters of interest. A different requirement that might be desirable in applied problems is to combine the accuracy of parameter estimation with the maximization of experimental output. The utility function considered here does this. We look at the implications of using this utility in deriving designs in the context of hierarchical linear models. In particular, designs for the one-way ANOVA and the straight-line models are obtained. New designs are determined analytically in some cases; in other cases we show that they can be computed by simple algorithms. An example is given to illustrate the elicitation of the parameter controlling the relative weight given to the two components of the utility.

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