Approximation of the Meta-Analytic-Predictive Prior Distribution in the One-Way Random Effects Model with Unknown Variance

In order to use historical data in the design of sample surveys with a Bayesian approach, the information from the historical data must be expressed as a prior distribution. Then, the best prior distribution for the parameter of interest is a predictive distribution. The density function of the predictive distribution generally is not available in an analytical form. From the perspective of practical use, Schmidli et al. (2014) proposed an approximation for the predictive distribution using a mixture of conjugate prior distributions. Their method relies on random numbers drawn from the predictive distribution. However, if the population distribution includes a nuisance parameter, their method becomes impractical. We propose a new approximation method that does not rely on these simulated numbers. Our approximation instead minimizes the mean squared error between the exact Bayes estimator and the one corresponding to the approximated predictive distribution.