Optimal Linear Opinion Pools

Consider a decision problem involving a group of m Bayesians in which each member reports his/her posterior distribution for some random variable θ. The individuals all share a common prior distribution for θ and a common loss function, but form their posterior distributions based on different data sets. A single distribution of θ must be chosen by combining the individual posterior distributions in some type of opinion pool. In this paper, the optimal pool is presented when the data observed by the different members of the group are conditionally independent given θ. When the data are not conditionally independent, the optimal weights to be used in a linear opinion pool are determined for problems involving quadratic loss functions and arbitrary distributions for θ and the data. Properties of the optimal procedure are developed and some examples are discussed.