  Most works on federated learning (FL) focus on the most common frequentist
formulation of learning whereby the goal is minimizing the global empirical
loss. Frequentist learning, however, is known to be problematic in the regime
of limited data as it fails to quantify epistemic uncertainty in prediction.
Bayesian learning provides a principled solution to this problem by shifting
the optimization domain to the space of distribution in the model parameters.
{\color{black}This paper proposes a novel mechanism for the efficient
implementation of Bayesian learning in wireless systems. Specifically, we focus
on a standard gradient-based Markov Chain Monte Carlo (MCMC) method, namely
Langevin Monte Carlo (LMC), and we introduce a novel protocol, termed Wireless
Federated LMC (WFLMC), that is able to repurpose channel noise for the double
role of seed randomness for MCMC sampling and of privacy preservation.} To this
end, based on the analysis of the Wasserstein distance between sample
distribution and global posterior distribution under privacy and power
constraints, we introduce a power allocation strategy as the solution of a
convex program. The analysis identifies distinct operating regimes in which the
performance of the system is power-limited, privacy-limited, or limited by the
requirement of MCMC sampling. Both analytical and simulation results
demonstrate that, if the channel noise is properly accounted for under suitable
conditions, it can be fully repurposed for both MCMC sampling and privacy
preservation, obtaining the same performance as in an ideal communication
setting that is not subject to privacy constraints.
