We consider a group of Bayesian agents who try to estimate a state of the
world θ through interaction on a social network. Each agent v
initially receives a private measurement of θ: a number Sv picked
from a Gaussian distribution with mean θ and standard deviation one.
Then, in each discrete time iteration, each reveals its estimate of θ 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⋅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 Our results extend those in an unpublished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.Comment: Added coauthor. Added proofs for fast convergence on trees and
distance transitive graphs. Also, now analyzing a notion of privac