Statistica Sinica 28 (2018), 505-525

SINGULAR PRIOR DISTRIBUTIONS AND

ILL-CONDITIONING IN BAYESIAN *D*-OPTIMAL

DESIGN FOR SEVERAL NONLINEAR MODELS

Timothy W. Waite

University of Manchester

Abstract: For Bayesian *D*-optimal design, we define a *singular prior distribution* for
the model parameters as 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 several multi-parameter generalized linear models,
we establish sufficient conditions for singularity of a prior distribution. For the
generalized linear models we also obtain sufficient conditions for non-singularity.
In the existing literature, weakly informative prior distributions are commonly recommended as a default choice for inference in logistic regression. Here it is shown
that some of the recommended prior distributions are singular, and hence should
not be used for Bayesian *D*-optimal design. Additionally, methods are developed
to derive and assess Bayesian *D*-efficient designs when numerical evaluation of the
objective function fails due to ill-conditioning, as often occurs for heavy-tailed prior
distributions. These numerical methods are illustrated for logistic regression.

Key words and phrases: Compartmental model, exponential decay model, generalized linear model, ill-conditioning, logistic regression.