Reaching Consensus with Imprecise Probabilities over a Network

This paper discusses the problem of a distributed network of agents attempting to agree on an imprecise probability over a network. Unique from other related work however, the agents must reach agreement while accounting for relative uncertainties in their respective probabilities. First, we assume that the agents only seek to agree to a centralized estimate of the probabilities, without accounting for observed transitions. We provide two methods by which such an agreement can occur which uses ideas from Dirichlet distributions. The first methods interprets the consensus problem as an aggregation of Dirichlet distributions of the neighboring agents. The second method uses ideas from Kalman Consensus to approximate this consensus using the mean and the variance of the Dirichlet distributions. A key results of this paper is that we show that when the agents are simultaneously actively observing state transitions and attempting to reach consensus on the probabilities, the agreement protocol can be insensitive to any new information, and agreement is not possible. Ideas from exponential fading are adopted to improve convergence and reach a consistent agreement.This research was funded in part under Air Force Grants # F49620-01-1-0453 and # FA9550-08-1-0086