Bayesian inference has widely acknowledged advantages in many problems, but it can also be unreliable when the model is misspecified. Bayesian modular inference is concerned with complex models which have been specified through a collection of coupled submodels, and is useful when there is misspecification in some of the submodels. The submodels are often called modules in the literature. 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, it may be hard to decide in what circumstances this ``cut posterior'' is preferred 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, we precisely formalize the bias-variance trade-off involved in semi-modular inference for the first time in the literature, using a framework of local model misspecification. 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 either the cut or full posterior on its own. The new method is demonstrated in a number of applications.

翻译：贝叶斯推断在许多问题中被广泛认为具有优势，但当模型设定错误时，其可靠性可能受到影响。贝叶斯模块化推断关注通过一组耦合子模型指定的复杂模型，当部分子模型存在设定错误时尤为有用。在文献中，这些子模型常被称为模块。切断反馈是一种广泛使用的贝叶斯模块化推断方法，可确保来自可疑模型组件的信息不会用于对正确设定模块中的参数进行推断。然而，在何种情况下这种"切断后验"优于精确后验可能难以判断。当模型设定错误不严重时，切断反馈可能在不显著减少估计偏差的情况下大幅增加贝叶斯后验推断的不确定性。这促使了半模块化推断方法的发展，该方法避免了切断反馈方法的二元切断特性。本研究首次在文献中利用局部模型设定错误框架精确形式化了半模块化推断中涉及的偏差-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，与单独使用切断后验或完整后验相比，该方法能在用户定义的损失函数下提供更准确的推断。该新方法在多个应用场景中得到了验证。