A Semidefinite Programming based approach for finding Bayesian optimal designs for nonlinear models

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 a 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-, Aor 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 discretizing schemes for the design space, different amount 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 Doptimal designs with two regressors for a logistic model and a two-variable generalized linear model with a gamma distributed response are discussed and some limitations of our approach are noted. AMS 2000 subject classifications: Primary 62K05, 90C22; secondary 65D32.