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In capital intensive semiconductor manufacturing processes it is often impractical to run large designed experiments and the amount of experimental data available is often not adequate to build sufficiently accurate statistical models or reliably estimating optimal conditions. This paper presents a new Bayesian predictive approach, referred to as the Bayesian adaptive design of experiments, for sequential design of experiments that aims to combine experimentation and optimization stages in order to start production more quickly with a small amount of process data. A dual control approach that simultaneously considers model estimation and optimization objectives is adopted and an adaptive Bayesian response surface model is used. It is shown that the optimal solution of the experimental settings can be determined either numerically for the case of a general second-order model or in analytical closed-form for the case of a first-order model. The effectiveness of the approach is illustrated with a simulation example and a real semiconductor process data taken from the literature. It is shown that by employing the proposed adaptive Bayesian approach one can simultaneously learn the process while not requiring excessive perturbations away from the target level and can achieve faster model estimation than central composite experimental designs. The learning weight used in the dual cost function allows one to tune the relative weights of learning and control goals depending on the uncertainty about the process model.