Quadrature Methods for Bayesian Optimal Design of Experiments With Nonnormal Prior Distributions

ABSTRACT

Many optimal experimental designs depend on one or more unknown model parameters. In such cases, it is common to use Bayesian optimal design procedures to seek designs that perform well over an entire prior distribution of the unknown model parameter(s). Generally, Bayesian optimal design procedures are viewed as computationally intensive. This is because they require numerical integration techniques to approximate the Bayesian optimality criterion at hand. The most common numerical integration technique involves pseudo Monte Carlo draws from the prior distribution(s). For a good approximation of the Bayesian optimality criterion, a large number of pseudo Monte Carlo draws is required. This results in long computation times. As an alternative to the pseudo Monte Carlo approach, we propose using computationally efficient Gaussian quadrature techniques. Since, for normal prior distributions, suitable quadrature techniques have already been used in the context of optimal experimental design, we focus on quadrature techniques for nonnormal prior distributions. Such prior distributions are appropriate for variance components, correlation coefficients, and any other parameters that are strictly positive or have upper and lower bounds. In this article, we demonstrate the added value of the quadrature techniques we advocate by means of the Bayesian D-optimality criterion in the context of split-plot experiments, but we want to stress that the techniques can be applied to other optimality criteria and other types of experimental designs as well. Supplementary materials for this article are available online.

KEYWORDS:

• Bayesian optimal design
• Beta distribution
• D-optimality
• Gamma distribution
• Log-normal distribution
• Numerical integration

Supplementary Materials

The supplementary materials contain 5 Matlab programs:

1.	GHquadrature.m: Gauss-Hermite quadrature for the lognormal prior distribution.
2.	GJbpquadrature.m: Gauss-Jacobi quadrature for the beta prime prior distribution.
3.	GJquadrature.m: Gauss-Jacobi quadrature for the beta prior distribution.
4.	GLquadrature.m: Gauss-Laguerre quadrature for the gamma prior distribution.
5.	GSWquadrature.m: Gauss-Stieltjes-Wigert quadrature for the lognormal prior distribution.

Acknowledgments

The authors thank Professor Walter Van Assche of the KU Leuven for his insights regarding the Stieltjes–Wigert polynomials and Gauss–Jacobi quadrature. The authors also thank the two referees for their constructive comments and suggestions.

The second author has received funding from the Universidad Carlos III de Madrid, the European Union’s Seventh Framework Programme for research, technological development and demonstration under grant agreement nr. 600371, el Ministerio de Economía y Competitividad (COFUND2013-40258), el Ministerio de Educación, Cultura y Deporte (CEI-15-17), and Banco Santander.