Distributed Bayesian Estimation in Sensor Networks: Consensus on Marginal Densities

TL;DR

The paper develops distributed Bayesian estimation algorithms for sensor networks by formulating estimation as a stochastic optimization in the space of probability densities. It introduces two methods: distributed SMD (DSMD) to fuse full-state pdfs and distributed marginal SMD (DMSMD) to estimate marginals over locally relevant variable subsets, both with almost-sure convergence guarantees. A Gaussian variational inference specialization provides tractable updates for non-linear likelihoods (e.g., LiDAR) and scalable marginal estimation. The results show that the methods achieve consensus and high estimation accuracy while substantially reducing storage and communication compared to full-state methods, with practical validation in relative localization and distributed mapping tasks.

Abstract

In this paper, we aim to design and analyze distributed Bayesian estimation algorithms for sensor networks. The challenges we address are to (i) derive a distributed provably-correct algorithm in the functional space of probability distributions over continuous variables, and (ii) leverage these results to obtain new distributed estimators restricted to subsets of variables observed by individual agents. This relates to applications such as cooperative localization and federated learning, where the data collected at any agent depends on a subset of all variables of interest. We present Bayesian density estimation algorithms using data from non-linear likelihoods at agents in centralized, distributed, and marginal distributed settings. After setting up a distributed estimation objective, we prove almost-sure convergence to the optimal set of pdfs at each agent. Then, we prove the same for a storage-aware algorithm estimating densities only over relevant variables at each agent. Finally, we present a Gaussian version of these algorithms and implement it in a mapping problem using variational inference to handle non-linear likelihood models associated with LiDAR sensing.

Figures (4)

• Figure 1: Trajectories of estimated node positions $\mu_{i,t}$ in an $8$ agent ring network with true positions shown as blue squares (top). Estimation error $\|\mu_{i,t}-\boldsymbol{x}_i\|$ over $1600$ time steps (bottom).
• Figure 2: Plots of the $500$-step average localization error, given by $1/n\sum_{i \in \mathcal{V}}\Vert \mu_{i,t} - \boldsymbol{x}_i\Vert$, using belief propagation, circular belief propagation, the proposed marginal estimation, and full state estimation algorithms in an $8$ node network. The comparisons span measurement noise variances $\Sigma_{ij} = b \mathbb{I}_2$ for $b \in \left\{1,2,5,10\right\}$ and network connectivities ranging from a line graph with $7$-edges to a $27$-edge fully connected one.
• Figure 3: Agent trajectories with training samples collected by agent $1$ with free and occupied points in blue and yellow respectively (left). Binary data in all training sets with gray free and labeled occupied points (center). Distinct and shared feature points in the likelihoods of agents $1$ and $3$ (right).
• Figure 4: Predicted classes by the agents over the verification set with gray free points and labeled occupied ones (left). Occupancy probability for points in verification sets for agents $1$ and $3$ (center). $L^1$ error over the verification set during the $100$k training steps.

Theorems & Definitions (44)

• Definition 1: Bregman divergence
• Definition 2
• Definition 3
• Proposition 1
• Definition 4
• Lemma 2: Pinsker's Lemma MSP:60
• Lemma 3
• Definition 5
• Definition 6
• Lemma 4: Gladyshev's Lemma BTP:87