Bayesian inference has widely acknowledged advantages in many problems, but it can also be unreliable if the model is misspecified. Bayesian modular inference is concerned with inference in complex models which have been specified through a collection of coupled sub-models. The sub-models are called modules in the literature, and they often arise from modeling different data sources, or from combining domain knowledge from different disciplines. When some modules are misspecified, cutting feedback is a widely used Bayesian modular inference method which ensures that information from suspect model components is not used in making inferences about parameters in correctly specified modules. However, in general settings it is difficult to decide when this ``cut posterior'' is preferable to the exact posterior. When misspecification is not severe, cutting feedback may increase the uncertainty in Bayesian posterior inference greatly without reducing estimation bias substantially. This motivates semi-modular inference methods, which avoid the binary cut of cutting feedback approaches. In this work, using a local model misspecification framework, we provide the first precise formulation of the the bias-variance trade-off that has motivated the literature on semi-modular inference. We then implement a mixture-based semi-modular inference approach, demonstrating theoretically that it delivers inferences that are more accurate, in terms of a user-defined loss function, than if either the cut or full posterior were used by themselves. The new method is demonstrated in a number of applications.

翻译：贝叶斯推断在许多问题中具有广泛认可的优势，但当模型存在错误设定时也可能不可靠。贝叶斯模块化推断关注的是通过一组耦合子模型指定的复杂模型中的推断问题。文献中将子模型称为模块，它们通常源于对不同数据源的建模，或来自不同学科领域知识的整合。当某些模块存在错误设定时，"切断反馈"是一种广泛使用的贝叶斯模块化推断方法，该方法确保来自可疑模型成分的信息不会被用于对正确指定模块中参数进行推断。然而在一般设定下，很难判断这种"切断后验"何时优于精确后验。当错误设定不严重时，切断反馈可能大幅增加贝叶斯后验推断的不确定性而无法显著降低估计偏差。这促使了半模块化推断方法的发展，这类方法避免了切断反馈方法的二元分割。本文利用局部模型错误设定框架，首次精确刻画了驱动半模块化推断文献的偏差-方差权衡。随后我们实现了一种基于混合的半模块化推断方法，从理论上证明该方法能根据用户自定义损失函数提供比单独使用切断后验或完整后验更准确的推断结果。新方法在多个应用中得到了验证。