This work considers Bayesian inference under misspecification for complex statistical models comprised of simpler submodels, referred to as modules, that are coupled together. Such ``multi-modular" models often arise when combining information from different data sources, where there is a module for each data source. When some of the modules are misspecified, the challenges of Bayesian inference under misspecification can sometimes be addressed by using ``cutting feedback" methods, which modify conventional Bayesian inference by limiting the influence of unreliable modules. Here we investigate cutting feedback methods in the context of generalized posterior distributions, which are built from arbitrary loss functions, and present novel findings on their behaviour. We make three main contributions. First, we describe how cutting feedback methods can be defined in the generalized Bayes setting, and discuss the appropriate scaling of the loss functions for different modules to each other and the prior. Second, we derive a novel result about the large sample behaviour of the posterior for a given module's parameters conditional on the parameters of other modules. This formally justifies the use of conditional Laplace approximations, which provide better approximations of conditional posterior distributions compared to conditional distributions from a Laplace approximation of the joint posterior. Our final contribution leverages the large sample approximations of our second contribution to provide convenient diagnostics for understanding the sensitivity of inference to the coupling of the modules, and to implement a new semi-modular posterior approach for conducting robust Bayesian modular inference. The usefulness of the methodology is illustrated in several benchmark examples from the literature on cut model inference.

翻译：本文考虑了在模型误设定情形下，针对由多个简单子模型（称为模块）耦合而成的复杂统计模型的贝叶斯推断。这类“多模块”模型常见于整合不同数据源信息时，每个数据源对应一个模块。当部分模块存在误设定时，通过使用“切割反馈”方法可以应对误设定下贝叶斯推断的挑战——该方法通过限制不可靠模块的影响来修改传统贝叶斯推断。本文在基于任意损失函数构建的广义后验分布框架下研究切割反馈方法，并揭示其行为的新发现。我们做出三项主要贡献：首先，描述了如何在广义贝叶斯设定中定义切割反馈方法，并讨论了损失函数在不同模块间及与先验之间的适当缩放准则；其次，推导了在给定其他模块参数条件下，特定模块参数后验分布大样本行为的新结果，这正式论证了条件拉普拉斯近似的合理性——与联合后验的拉普拉斯近似得到的条件分布相比，该近似能更精确地逼近条件后验分布；第三项贡献基于第二项贡献的大样本近似，为理解推断对模块耦合的敏感性提供了便捷的诊断工具，并实现了一种新的半模块化后验方法以进行稳健的贝叶斯模块化推断。通过文献中关于切割模型推断的多个基准示例，验证了该方法的实用性。