[Paper Review] Distributed Detection via Bayesian Updates and Consensus

Qipeng Liu, Jiuhua Zhao+1 more|arXiv (Cornell University)|Dec 17, 2014

Distributed Sensor Networks and Detection Algorithms14 references3 citations

TL;DR

This paper proposes and compares six distributed detection algorithms that combine Bayesian updates with consensus protocols—using either arithmetic or geometric averaging—across agents in a network. The key finding is that geometric averaging consistently accelerates convergence to the true state, while communication delays and signal structure diversity significantly impact detection speed.

ABSTRACT

In this paper, we discuss a class of distributed detection algorithms which can be viewed as implementations of Bayes' law in distributed settings. Some of the algorithms are proposed in the literature most recently, and others are first developed in this paper. The common feature of these algorithms is that they all combine (i) certain kinds of consensus protocols with (ii) Bayesian updates. They are different mainly in the aspect of the type of consensus protocol and the order of the two operations. After discussing their similarities and differences, we compare these distributed algorithms by numerical examples. We focus on the rate at which these algorithms detect the underlying true state of an object. We find that (a) The algorithms with consensus via geometric average is more efficient than that via arithmetic average; (b) The order of consensus aggregation and Bayesian update does not apparently influence the performance of the algorithms; (c) The existence of communication delay dramatically slows down the rate of convergence; (d) More communication between agents with different signal structures improves the rate of convergence.

Motivation & Objective

• To analyze and compare a class of distributed detection algorithms that integrate Bayesian updating with consensus protocols in networked systems.
• To investigate how variations in consensus type (arithmetic vs. geometric), operation order (update vs. consensus), and communication delay affect detection efficiency.
• To provide empirical insights into designing faster, more reliable distributed detection systems through numerical simulations.
• To explore whether theoretical conditions for convergence can be relaxed under different network and signal structures.
• To compare the performance of algorithms under different agent clustering and signal structure configurations.

Proposed method

• Model a networked system as a directed graph where agents exchange beliefs through weighted communication links with self-weights summing to one.
• Implement Bayesian updates using private signal likelihoods to compute posterior beliefs based on observed signals.
• Apply consensus protocols via arithmetic averaging (e.g., BLiA, LiAB) or geometric averaging (e.g., BLoA, BLoAD) to aggregate neighbor beliefs.
• Introduce delayed communication by using prior beliefs instead of current posterior beliefs in consensus steps (e.g., BLiAD, BLoAD).
• Use a 20-agent network with three possible states, where only the global structure identifies the true state (θ₃), and agents have asymmetric signal structures.
• Simulate belief evolution over time and measure convergence speed as the number of iterations until all agents’ confidence in θ₃ exceeds 0.999.

Experimental results

Research questions

• RQ1 How does the choice between arithmetic and geometric averaging in consensus protocols affect the convergence rate of distributed detection?
• RQ2 Does the order of Bayesian update and consensus aggregation significantly influence detection performance?
• RQ3 How do communication delays—using outdated neighbor beliefs—affect the speed of convergence to the true state?
• RQ4 To what extent does the heterogeneity of signal structures among agents impact the rate of collective detection?
• RQ5 Can the theoretical convergence conditions for distributed detection be relaxed under time-varying or clustered network topologies?

Key findings

• Algorithms using geometric averaging (e.g., BLoA, BLoAD, LoAB) converge significantly faster than those using arithmetic averaging (e.g., BLiA, BLiAD, LiAB), with geometric averaging showing superior detection efficiency.
• The order of Bayesian update and consensus aggregation has no apparent influence on detection performance, indicating robustness to operation sequence in the algorithmic design.
• Communication delays—where agents use outdated prior beliefs in consensus—dramatically slow down convergence, as seen in BLiAD and BLoAD algorithms.
• Agents with diverse signal structures (e.g., V₁ and V₂ with different likelihood mappings) improve detection speed when mixed spatially, compared to clustered configurations.
• The lower bound on detection rate for BLoA exceeds the upper bound for BLiAD, confirming that geometric consensus is theoretically more efficient than delayed arithmetic consensus.
• In simulations, mixing agents from different signal structure groups (V₁ and V₂) accelerates convergence compared to clustering, highlighting the benefit of structural diversity in information sharing.