Hierarchical Fully Probabilistic Design for Deliberator-Based Merging in Multiple Participant Systems

Abstract

We address the problem of merging probabilistic knowledge in a centralized (nonflat) multiple participant system. Each participant independently reports a known distribution of a common quantity of interest to an extra participant, called the deliberator. The deliberator’s task is the optimal design of a compromise distribution, which seeks to minimize the expected loss imposed on each of the participants. We formulate this task as one of Bayesian decision-making, and solve it using fully probabilistic design. Solutions are currently available for discrete distributions only. In this paper, we provide two novel contributions. First, the deliberator’s unknown distribution is parametric on a continuous support. Second, the scheme adopts a hierarchical model for the deliberator, optimizing the deliberator’s parameter prior. This allows quantification of the deliberator’s uncertainty in their model choice. We formulate conditions under which a conjugate design is achieved, and provide an example where the design is analytically tractable. Extensive simulation-based evidence is reported, pointing to the significantly enhanced performance of our framework over logarithmic pools.