  Conventional frequentist learning, as assumed by existing federated learning
protocols, is limited in its ability to quantify uncertainty, incorporate prior
knowledge, guide active learning, and enable continual learning. Bayesian
learning provides a principled approach to address all these limitations, at
the cost of an increase in computational complexity. This paper studies
distributed Bayesian learning in a wireless data center setting encompassing a
central server and multiple distributed workers. Prior work on wireless
distributed learning has focused exclusively on frequentist learning, and has
introduced the idea of leveraging uncoded transmission to enable "over-the-air"
computing. Unlike frequentist learning, Bayesian learning aims at evaluating
approximations or samples from a global posterior distribution in the model
parameter space. This work investigates for the first time the design of
distributed one-shot, or "embarrassingly parallel", Bayesian learning protocols
in wireless data centers via consensus Monte Carlo (CMC). Uncoded transmission
is introduced not only as a way to implement "over-the-air" computing, but also
as a mechanism to deploy channel-driven MC sampling: Rather than treating
channel noise as a nuisance to be mitigated, channel-driven sampling utilizes
channel noise as an integral part of the MC sampling process. A simple wireless
CMC scheme is first proposed that is asymptotically optimal under Gaussian
local posteriors. Then, for arbitrary local posteriors, a variational
optimization strategy is introduced. Simulation results demonstrate that, if
properly accounted for, channel noise can indeed contribute to MC sampling and
does not necessarily decrease the accuracy level.
