Singular prior distributions in Bayesian D-optimal design for nonlinear models

Abstract

For Bayesian D-optimal design, we define a singular prior distribution to be a prior distribution such that the determinant of the Fisher information matrix has a prior geometric mean of zero for all designs. For such a prior distribution, the Bayesian D-optimality criterion fails to select a design. For the exponential decay model, we characterize singularity of the prior distribution in terms of the expectations of a few elementary transformations of the parameter. For a compartmental model and multi-parameter logistic regression we establish sufficient conditions for singularity of a prior distribution. For logistic regression we also obtain sufficient conditions for non-singularity. The results are applied to show that the weakly informative prior distribution proposed as a default for inference by Gelman, Jakulin, Pittau and Su (2008) should not be used for Bayesian D-optimal design. Additionally, we develop methods to derive and assess Bayesian D-efficient designs for logistic regression when numerical evaluation of the objective function fails due to ill-conditioning.

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