Merging experts' opinions: A Bayesian hierarchical model with mixture of prior distributions

In this paper, a general approach is proposed to address a full Bayesian analysis for the class of quadratic natural exponential families in the presence of several expert sources of prior information. By expressing the opinion of each expert as a conjugate prior distribution, a mixture model is used by the decision maker to arrive at a consensus of the sources. A hyperprior distribution on the mixing parameters is considered and a procedure based on the expected Kullback-Leibler divergence is proposed to analytically calculate the hyperparameter values. Next, the experts' prior beliefs are calibrated with respect to the combined posterior belief over the quantity of interest by using expected Kullback-Leibler divergences, which are estimated with a computationally low-cost method. Finally, it is remarkable that the proposed approach can be easily applied in practice, as it is shown with an application.