Distributed Bayesian Estimation in Sensor Networks: Consensus on Marginal Densities

In this paper, we design and analyze distributed Bayesian estimation algorithms for sensor networks. We consider estimation problems, such as cooperative localization and federated learning, where the data collected at any agent depends on a subset of all variables of interest. We provide a unified formulation of centralized, distributed and marginal probabilistic estimation as a Bayesian density estimation problem using data from non-linear likelihoods at agent. We develop distributed estimation algorithms based on stochastic mirror descent with appropriate regularization to enforce distributed or marginal density constraints. We prove almost-sure convergence to the optimal set of probabilities at each agent in both the distributed and marginal settings. Finally, we present Gaussian density versions of these algorithms and compare them to belief propagation variants in a node localization problem with relative position measurements. We also demonstrate our algorithms in a multi-agent mapping problem using LiDAR data.