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. This paper studies for the first time Bayesian FL in wireless systems by proposing and analyzing a gradient-based Markov Chain Monte Carlo (MCMC) method -- Wireless Federated Langevin Monte Carlo (WFLMC). The key idea of this work is to repurpose channel noise for the double role of seed randomness for MCMC sampling and of privacy-preserving mechanism. 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.
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