Inference on exponential families with mixture of prior distributions

A Bayesian analysis of the natural exponential families with quadratic variance function when there are several sources of prior information is considered. The belief of each source is expressed as a conjugate prior distribution. Then, a mixture of them is considered to represent a consensus of the sources. A unified framework considering unknown weights is presented. Firstly, a general procedure based on Kullback–Leibler (K–L) distance to obtain the weights is proposed. The main advantage is that the weights can be analytically calculated. In addition, expressions that allow a direct implementation for these families are shown. Secondly, the experts’ prior beliefs are calibrated with respect to the combined posterior belief by using K–L distances. A straightforward Monte Carlo-based approach to estimate these distances is proposed. Finally, two illustrative examples are presented to show the ease of application of the proposed technique, as well as its usefulness in a Bayesian framework.