Reaching Consensus with Imprecise Probabilities over a Network

Abstract

Information consensus in sensor networks has received much attention due to its numerous applications in distributed decision making. This paper discusses the problem of a distributed group of agents coming to agreement on a probability vector over a network, such as would be required in a decentralized estimation of state transition probabilities or agreement on a probabilistic search map. Unique from other recent consensus literature, however, the agents in this problem must reach agreement while accounting for the uncertainties in their respective probabilities, which are formulated according to generally non-Gaussian distributions. The first part of this paper considers the problem in which the agents seek agreement to the centralized Bayesian estimate of the probabilities, which is accomplished using consensus on hyperparameters. The second part shows that the new hyperparameter consensus methodology can ensure convergence to the centralized estimate even while measurements of a static process are occurring concurrently with the consensus algorithm. A machine repair example is used to illustrate the advantages of hyperparameter consensus over conventional consensus approaches.

A hyperparameter-based approach for consensus under uncertainties

“…Theorem 3.3 (Convergence with Measurements): A group of N agents will come to an asymptotic agreement on the Bayesian fused distribution on an uncertain parameter, θ, in the presence of global common information and a finite number of concurrent measurements, by running the consensus algorithm defined by (11) through (14) , with the following properties holding true:…”

“…If the measurement is shared among all agents, such that z i [κ] = z ⋆ [κ] ∀ i, then each agent must update their hyperparameters as in (14).…”

“…These algorithms give rise to a consensus result that is influenced more heavily by agents with smaller covariance (therefore higher certainty) in their estimates. Unfortunately, many stochastic systems do not satisfy the assumption of a linear system with Gaussian noises, and applying these Kalman filter-based consensus methods to other distributions can produce a steady-state estimate that is biased away from the Bayesian estimate obtained by using the true form of the distribution [14].…”

“…[14] derived a method to permit a consensus on the Bayesian aggregation of multiple Dirichlet distributions, and demonstrated its use in distributed estimation of transition probabilities in a multi-agent learning problem. This paper utilizes the same concepts of pseudo-measurement updates and conjugacy of probability distributions [22] to extend the Bayesian fusion results obtained in [14] to a more general class of distributions.…”

“…Recent results have shown that a combination of traditional consensus-based communication protocols with data fusion information updates can achieve scalable, representative information fusion results without requiring complex channel filters or specific network topologies [14], [19]- [21]. In particular, [19], [20] utilized dynamic-average consensus filters to achieve an approximate distributed Kalman filter, while [21] implemented a linear consensus protocol on the parameters of the information form of the Kalman filter.…”