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 in order 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 non-normal prior distributions. Such prior distributions are appropriate for variance components, correlation coefficients and any other parameters that are strictly positive or have 1 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.

A Metaheuristic Adaptive Cubature Based Algorithm to Find Bayesian Optimal Designs for Nonlinear Models

“…Similarly, the Gauss-Hermite formula, which is based on the Gauss-Hermite polynomials, can be applied when we have a normal prior distribution. For more details, see K. E. and Goos and Mylona (2018). The ICA algorithm is very flexible and can also be combined with the Gaussian quadrature formulas.…”

“…The MC method uses a sequence of random draws from the prior distribution to approximate the integral and it usually requires a large number of function evaluations to provide accurate estimates of the integrals. Quadrature formulas are alternative deterministic methods that approximate the integrals by a weighted sum of the integrand values at specific points, called nodes; for a review, see Goos and Mylona (2018). Although the quadrature techniques are fast, they usually do not provide estimation errors and their accuracy depends on the prior distribution and properties of the integrand.…”

D-Optimal Design for a Causal Structure for Completely Randomized and Random Blocked Experiments

“…When there is no prior data available, a second approach is a Bayesian approach which assumes a prior distribution for the endogenous variables, which is similar to the approach that is used by Mylona et al [25] and Goos and Mylona [38] to address the variance components optimal design for blocked or split-plot experiments. is approach allows for the uncertainty of the endogenous parameters.…”

Optimal Blocked and Split-Plot Designs Ensuring Precise Pure-Error Estimation of the Variance Components

“…For the examples we discuss below, using this alternative does not lead to qualitatively dierent results than using the lognormal prior for η. Goos and Mylona (2018) provide a tutorial about computationally ecient quadrature approaches to deal with various kinds of non-normal prior distributions.…”