Distributed estimation using Bayesian consensus filtering

Abstract

Abstract-We present the Bayesian consensus filter (BCF) for tracking a moving target using a networked group of sensing agents and achieving consensus on the best estimate of the probability distributions of the target's states. Our BCF framework can incorporate nonlinear target dynamic models, heterogeneous nonlinear measurement models, non-Gaussian uncertainties, and higher-order moments of the locally estimated posterior probability distribution of the target's states obtained using Bayesian filters. If the agents combine their estimated posterior probability distributions using a logarithmic opinion pool, then the sum of Kullback-Leibler divergences between the consensual probability distribution and the local posterior probability distributions is minimized. Rigorous stability and convergence results for the proposed BCF algorithm with single or multiple consensus loops are presented. Communication of probability distributions and computational methods for implementing the BCF algorithm are discussed along with a numerical example.

Distributed Bayesian filtering using logarithmic opinion pool for dynamic sensor networks

“…Lemma 4 [16,17] The pdf P KL k ∈ Φ(X ) that globally minimizes the sum of KL divergences with the pdfs P i k for all agents is given by:…”

“…In contrast, we present a rigorous proof technique, which was first introduced in our prior work [17,18], for the LogOP scheme that is applicable for general probability distributions. This paper is organized as follows.…”

“…Combining probability distributions within the exponential family (i.e., probability distributions that can be expressed as exponential functions) is discussed in [13,14]. In the distributed estimation algorithm presented in [16] as well as in our prior work [17,18], the agents combine their local posterior probability distributions using the consensus algorithm, where the multiple consensus loops within each time step are executed much faster than the original time steps of the Bayesian filter. Moreover, [16,17,18] show that each agent's estimated probability distribution of the target's states converges around the pdf that minimizes the sum of Kullback-Leibler (KL) divergences from all the posterior probability distributions of the target's states.…”

“…In the distributed estimation algorithm presented in [16] as well as in our prior work [17,18], the agents combine their local posterior probability distributions using the consensus algorithm, where the multiple consensus loops within each time step are executed much faster than the original time steps of the Bayesian filter. Moreover, [16,17,18] show that each agent's estimated probability distribution of the target's states converges around the pdf that minimizes the sum of Kullback-Leibler (KL) divergences from all the posterior probability distributions of the target's states. Similar algorithms for combining local likelihood functions using the consensus algorithm are proposed in [14,15].…”

“…The third category of algorithms estimates the posterior probability distribution of the states of the target [11,12,13,14,15,16,17,18]. This category forms the most general class of distributed estimation algorithms because these algorithms can be used for estimation over continuous domains, and can incorporate nonlinear target dynamics, heterogeneous nonlinear measure-ment models, and non-Gaussian uncertainties.…”