Efficient Bayesian Learning in Social Networks with Gaussian Estimators

TL;DR

This paper presents an efficient Bayesian learning process for social networks where agents iteratively update their beliefs about a hidden state using Gaussian estimators, ensuring quick convergence, privacy preservation, and computational simplicity.

Contribution

It introduces a computationally efficient Bayesian updating method for agents in social networks that guarantees fast convergence and privacy preservation, extending previous work.

Findings

Agents converge to the optimal belief after at most 2N·D steps.

On trees and distance transitive-graphs, convergence occurs after D steps.

The process preserves privacy while aggregating information efficiently.

Abstract

We consider a group of Bayesian agents who try to estimate a state of the world $\theta$ through interaction on a social network. Each agent $v$ initially receives a private measurement of $\theta$: a number $S_v$ picked from a Gaussian distribution with mean $\theta$ and standard deviation one. Then, in each discrete time iteration, each reveals its estimate of $\theta$ to its neighbors, and, observing its neighbors' actions, updates its belief using Bayes' Law. This process aggregates information efficiently, in the sense that all the agents converge to the belief that they would have, had they access to all the private measurements. We show that this process is computationally efficient, so that each agent's calculation can be easily carried out. We also show that on any graph the process converges after at most $2N \cdot D$ steps, where $N$ is the number of agents and $D$ is the diameter of the network. Finally, we show that on trees and on distance transitive-graphs the process converges after $D$ steps, and that it preserves privacy, so that agents learn very little about the private signal of most other agents, despite the efficient aggregation of information. Our results extend those in an unpublished manuscript of the first and last authors.