Log-Linear Pool to Combine Prior Distributions: A Suggestion for a Calibration-Based Approach

An important issue involved in group decision making is the suitable aggregation of experts’ beliefs about a parameter of interest. Two widely used combination methods are linear and log-linear pools. Yet, a problem arises when the weights have to be selected. This paper provides a general decision-based procedure to obtain the weights in a log-linear pooled prior distribution. The process is based on Kullback-Leibler divergence, which is used as a calibration tool. No information about the parameter of interest is considered before dealing with the experts’ beliefs. Then, a pooled prior distribution is achieved, for which the expected calibration is the best one in the Kullback-Leibler sense. In the absence of other information available to the decision-maker prior to getting experimental data, the methodology generally leads to selection of the most diffuse pooled prior. In most cases, a problem arises from the marginal distribution related to the noninformative prior distribution since it is improper. In these cases, an alternative procedure is proposed. Finally, two applications show how the proposed techniques can be easily applied in practice.