title: Finding Bayesian Optimal Designs for Nonlinear Models: A Semidefinite Programming-Based Approach.
abstract: This paper uses semidefinite programming (SDP) to construct Bayesian optimal design for nonlinear regression models. The setup here extends the formulation of the optimal designs problem as an SDP problem from linear to nonlinear models. Gaussian quadrature formulas (GQF) are used to compute the expectation in the Bayesian design criterion, such as D-, A- or E-optimality. As an illustrative example, we demonstrate the approach using the power-logistic model and compare results in the literature. Additionally, we investigate how the optimal design is impacted by different discretising schemes for the design space, different amounts of uncertainty in the parameter values, different choices of GQF and different prior distributions for the vector of model parameters, including normal priors with and without correlated components. Further applications to find Bayesian D-optimal designs with two regressors for a logistic model and a two-variable generalised linear model with a gamma distributed response are discussed, and some limitations of our approach are noted.
authors.affiliation: Department of Chemical and Biological Engineering, ISEC, Polytechnic Institute of Coimbra, R. Pedro Nunes, 3030-199 Coimbra, Portugal ; GEPSI, CIEPQPF, Department of Chemical Engineering, University of Coimbra, R. Sílvio Lima, Pólo II, 3030-790 Coimbra, Portugal.
authors.forename: Belmiro P M
authors.initials: BP
authors.lastname: Duarte
authors.affiliation: Department of Biostatistics, Fielding School of Public Health, UCLA, 10833 Le Conte Ave., Los Angeles, CA 90095-1772, USA.
authors.forename: Weng Kee
authors.initials: WK
authors.lastname: Wong
keywords: Approximate designs; Gaussian quadrature formulas; nonlinear models; semidefinite programming