  Motivated by the need to analyze large, decentralized datasets, distributed
Bayesian inference has become a critical research area across multiple fields,
including statistics, electrical engineering, and economics. This paper
establishes Frequentist properties, such as posterior consistency, asymptotic
normality, and posterior contraction rates, for the distributed (non-)Bayes
Inference problem among agents connected via a communication network. Our
results show that, under appropriate assumptions on the communication graph,
distributed Bayesian inference retains parametric efficiency while enhancing
robustness in uncertainty quantification. We also explore the trade-off between
statistical efficiency and communication efficiency by examining how the design
and size of the communication graph impact the posterior contraction rate.
Furthermore, We extend our analysis to time-varying graphs and apply our
results to exponential family models, distributed logistic regression, and
decentralized detection models.
