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：本文考虑在模型误设定情形下，针对由多个较简单子模型（称为模块）耦合而成的复杂统计模型进行贝叶斯推断。这类"多模块"模型通常出现在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，可通过使用"反馈切断"方法——通过限制不可靠模块的影响来修正传统贝叶斯推断——应对误设定下的贝叶斯推断挑战。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反馈切断方法，并揭示其行为的新发现。我们做出三项主要贡献：首先，描述如何在广义贝叶斯框架中定义反馈切断方法，并讨论各模块损失函数的合理尺度化及其与先验的配比关系；其次，推导出关于给定其他模块参数条件下特定模块参数后验的大样本行为的新结论，这为使用条件拉普拉斯近似提供了正式依据——相比联合后验的拉普拉斯近似导出的条件分布，该方法能更精确地逼近条件后验分布；最后，基于第二项贡献的大样本近似结果，提出便捷的诊断工具以理解推断对模块耦合的敏感性，并实现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。通过在切断模型推断文献中的多个基准示例验证了该方法的实用性。