Bayesian Poisson models for the graphical combination of dependent expert information

A supra-Bayesian (SB) wants to combine the information from a group of k experts to produce her distribution of a probability θ. Each expert gives his counts of what he thinks are the numbers of successes and failures in a sequence of independent trials, each with probability θ of success. These counts, used as a surrogate for each expert's own individual probability assessment (together with his associated level of confidence in his estimate), allow the SB to build various plausible conjugate models. Such models reflect her beliefs about the reliability of different experts and take account of different possible patterns of overlap of information between them. Corresponding combination rules are then obtained and compared with other more established rules and their properties examined.