  The recent development of scalable Bayesian inference methods has renewed
interest in the adoption of Bayesian learning as an alternative to conventional
frequentist learning that offers improved model calibration via uncertainty
quantification. Recently, federated averaging Langevin dynamics (FALD) was
introduced as a variant of federated averaging that can efficiently implement
distributed Bayesian learning in the presence of noiseless communications. In
this paper, we propose wireless FALD (WFALD), a novel protocol that realizes
FALD in wireless systems by integrating over-the-air computation and
channel-driven sampling for Monte Carlo updates. Unlike prior work on wireless
Bayesian learning, WFALD enables (\emph{i}) multiple local updates between
communication rounds; and (\emph{ii}) stochastic gradients computed by
mini-batch. A convergence analysis is presented in terms of the 2-Wasserstein
distance between the samples produced by WFALD and the targeted global
posterior distribution. Analysis and experiments show that, when the
signal-to-noise ratio is sufficiently large, channel noise can be fully
repurposed for Monte Carlo sampling, thus entailing no loss in performance.
