A Bayesian approach to linear calibration design problem

Based on linear models, the inference about the true measurement and the optimal designs for the calibration experiment are considered via Bayesian statistical decision analysis. The posterior distribution of given the observation and the calibration experiment is obtained with normal priors for and for the model parameters . This posterior distribution is not in the form of any known distributions, which leads to the use of a numerical integration or an approximation for the calculation of the overall expected loss. The general structure of the expected loss function is characterized in the form of a conjecture. The near-optimal design is obtained through the approximation of the conditional covariance matrix of the joint distribution of . Numerical results for the univariate case are given to demonstrate the conjecture and to evaluate the approximation.