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.

翻译：受大规模分散式数据分析需求的驱动，分布式贝叶斯推断已成为统计学、电气工程和经济学等多个领域的关键研究方向。本文针对通过通信网络连接的智能体之间的分布式（非）贝叶斯推断问题，建立了频率性质，包括后验一致性、渐近正态性及后验收缩速率。研究结果表明，在适当的通信图假设下，分布式贝叶斯推断在保持参数效率的同时，增强了不确定性量化的鲁棒性。通过考察通信图的结构与规模如何影响后验收缩速率，我们进一步探讨了统计效率与通信效率之间的权衡关系。此外，我们将分析拓展至时变图场景，并将所得结果应用于指数族模型、分布式逻辑回归及分散式检测模型。